# Effect of dipole size fluctuations on diffractive photo-production of vector
mesons

Jean-Paul Blaizot  Institut de Physique Théorique, Université Paris Saclay,
CEA, CNRS, F-91191 Gif-sur-Yvette, France<EMAIL_ADDRESS>Marco
Claudio Traini  INFN - TIFPA, Via Sommarive 14, I-38123 Trento-Povo, Italy
Dipartimento di Fisica, Università degli Studi di Trento, Via Sommarive 14,
I-38123 Trento-Povo, Italy

###### Abstract

We consider the diffractive photo-production of vector mesons on a proton, in
the dipole model. We take into account the effect of the fluctuations of the
the dipole size, whose magnitude is controlled by the overlap between the
photon and the vector meson wave functions. Our predictions for the incoherent
diffractive cross section, obtained within the Impact Parameter Saturation
Model (IPSat), are shown to be in excellent agreement with the HERA data on
$J/\Psi$ photo-production, down to very low momentum transfer, where the
dipole size fluctuations are shown to play an essential role. This study
complements, without introducing any additional parameter, previous treatments
of incoherent diffractive processes in terms of fluctuations of the proton
shape, by adding another source of “geometrical” fluctuations, namely those
coming from the splitting of the photon into color dipoles of various sizes.

##  I  Introduction

Deep inelastic scattering of leptons on hadrons is a tool of choice to obtain
information on the structure of hadrons, or nuclei, in terms of partons. At
high energy, and in an appropriate frame, it is often convenient to picture
the interaction as a two-step process, whereby the virtual photon exchanged
between the lepton and a hadron first splits into a quark-antiquark dipole,
which later interacts strongly with the hadron. When the interaction of the
dipole with the hadrons involves partons that carry a very small fraction $x$
of the hadron momentum, such an approach gives access to the gluon density in
the hadron and its possible saturation property Mueller (1990). This picture
is at the basis of the color dipole model Nikolaev and Zakharov (1991);
Mueller (1994), which exists in various incarnations (see e.g. Golec-Biernat
and Wusthoff (1998, 1999); Kowalski and Teaney (2003); Bartels  _et al._
(2002); Kowalski  _et al._ (2006)).

Dipole models are also useful to understand diffractive events, which
represent a large fraction of the electron-hadron cross section at high energy
(for a recent review see Frankfurt  _et al._ (2022)). Many studies have been
carried out within the last two decades using such models, with in particular
the goal of getting evidence for saturation effects in the diffractive
production of different final states Golec-Biernat and Wusthoff (1998, 1999);
Kowalski and Teaney (2003). The vector meson production off nucleons and
nuclei Ivanov  _et al._ (2006) is a particularly clean process since the final
state is unambiguously identified by a large rapidity gap. Such processes are
strongly related to the underlying QCD dynamics and are sensitive to the gluon
content of the target Kovchegov and Levin (2012). Among the many existing
models, a popular one is the impact parameter dependent saturation model
(IPsat) Bartels  _et al._ (2002); Kowalski and Teaney (2003); Kowalski  _et
al._ (2006); Rezaeian  _et al._ (2013); Cepila  _et al._ (2017) whose basic
parameters are determined through fits to available high precision data of
deep inelastic scattering of electrons on protons as measured at HERA H1, ZEUS
collaborations  _et al._ (2010, 2015) (see ref. Mäntysaari and Zurita (2018)
for a recent review). This is the model that we shall mainly use in this work.

However, our main interest in this paper is not the physics of saturation, but
rather that of fluctuations that dominate the diffractive cross section at
moderate momentum transfer. In this context, incoherent diffraction plays a
prominent role because it is sensitive to the transverse fluctuations of the
gluon field in the target. To study these fluctuations, we shall rely on the
formalism of diffraction eigenstates Good and Walker (1960); Miettinen and
Pumplin (1978). The states of the dipole, defined by the coordinates of the
quark and antiquark provide an example of such eigenstates. This is because,
when the dipole picture is a valid one, the lifetime of the dipole is long as
compared to the size of the hadron, and the dipole coordinates remain frozen
during the scattering. The coordinates of the sources of the color field of
the target, also frozen during the collision, provide another example of
diffraction eigenstate Miettinen and Pumplin (1978). In this paper we follow
the phenomenological approach initiated in Ref. Mäntysaari and Schenke (2016a)
and assume that a dominant source of fluctuations at moderate momentum
transfer is that due to the random location of the valence quarks in each
collision. Each valence quark is carrying its cloud of gluons, and the
fluctuation of their locations entails a corresponding fluctuation of the
gluon field in the proton. We shall stick to this simple picture, being aware
of the existence of other models where this relation between the gluon field
and the valence quarks is relaxed, so-called “hot spot” models which can be
extended to momentum transfers higher than those we shall consider in this
paper (see e.g. Demirci  _et al._ (2022); Kumar and Toll (2021); Albacete and
Soto-Ontoso (2017) and for a review Mäntysaari (2020)).

The main novelty of the present work concerns the treatment of the
fluctuations of the dipole size. Such fluctuations result, event-by-event,
from the quantum mechanical process of the photon splitting. Thus one may
attach a probability to the splitting, given in principle by the photon wave
function. In the case of vector meson production, a subtlety arises due to the
fact that what plays the role of a probability is the overlap of the photon
wave function with that of the vector meson Munier  _et al._ (2001). We shall
elaborate on this feature and propose a treatment of geometrical fluctuations
that include those of the dipole size. We shall see that such fluctuations
account correctly for the incoherent cross section in the region of small
momentum transfer where the fluctuations of the proton shape die out. These
dipole size fluctuations, which translate into fluctuations of the dipole
cross section, bear some analogy with the fluctuating cross sections
introduced in Refs. Blaettel  _et al._ (1993a, b). They could also be
interpreted as fluctuations of the saturation momentum Mäntysaari and Schenke
(2016a). However the simple treatment of dipole size fluctuations suggested in
the present paper appears more natural and does not require any new ingredient
beyond those already fixed in the model. Note finally that we shall ignore
other types of fluctuations, namely those corresponding to gluon emissions
Marquet (2007), intrinsic fluctuations of the color field within the hot spots
Schlichting and Schenke (2014), as well as the effects of small-$x$ evolution
Liou  _et al._ (2017).

The outline of the paper is as follows. In the next section we briefly review
the formalism commonly used to calculate the vector meson production in the
dipole model. Elaborating on the formalism of diffraction eigenstates Good and
Walker (1960); Miettinen and Pumplin (1978), we recall in Sect. II.2 how
fluctuations of the locations of the sources of the gluon field contribute to
the incoherent diffractive cross section. We give a simple argument, based on
the relation between these geometrical fluctuations and the density-density
correlation function, why the contribution of these fluctuations to the
incoherent cross section vanishes at small momentum transfer. In Sect. II.3 we
develop our suggestion for estimating the magnitude of the fluctuations of the
dipole size. The following section, Sec. III, provides details on the
numerical calculations and discusses our results and their comparison with
HERA data. The last section summarizes our conclusions. Various appendices
collect technical details on the calculation.

##  II Vector meson production in the dipole model

The amplitude ${\cal A}_{T,L}^{\gamma^{*}p\to
Vp^{\prime}}({x_{\\!I\\!\\!P}},Q^{2},\mbox{\boldmath$\Delta$})$ for
diffractive vector meson production on a proton assumes the form Kowalski  _et
al._ (2006)

$\displaystyle{\cal A}_{T,L}^{\gamma^{*}p\to
Vp^{\prime}}({x_{\\!I\\!\\!P}},Q^{2},\mbox{\boldmath$\Delta$})$ (1)
$\displaystyle=$ $\displaystyle i\int d^{2}{\bf r}\int_{0}^{1}\frac{{\rm
d}z}{4\pi}\,\Psi_{V}^{*}(z,r,Q^{2}){\cal
A}_{q\bar{q}}({x_{\\!I\\!\\!P}},Q^{2},{\bm{r}},\mbox{\boldmath$\Delta$})\Psi_{T,L}(r,z,Q^{2}).$

The subscripts $T$ and $L$ refer to the transverse and longitudinal
polarizations of the exchanged virtual photon $\gamma^{*}$,
${x_{\\!I\\!\\!P}}=(P-P^{\prime})\cdot q/(P\cdot q)$ is the fraction of the
longitudinal momentum of the proton involved in the exchange, and the total
momentum transfer (squared) is $t=-(P^{\prime}-P)^{2}$ with $P$ and
$P^{\prime}$ the initial and final proton four-momenta. The photon-proton
scattering is characterized by a total center-of-mass-energy squared
$W^{2}=(P+q)^{2}$, ($Q^{2}=-q^{2}$). Finally
${\mbox{\boldmath$\Delta$}}=(P^{\prime}-P)_{\perp}$ is the transverse momentum
transfer. The normalisation of the amplitude is such that the coherent cross
section is given by Munier  _et al._ (2001)

$\displaystyle{d\sigma_{T,L}^{\gamma^{*}p\to Vp^{\prime}}\over
dt}={(1+\beta^{2})\over 16\pi}\left|{\cal A}_{T,L}^{\gamma^{*}p\to
Vp}({x_{\\!I\\!\\!P}},Q^{2},\mbox{\boldmath$\Delta$})\right|^{2}.$ (2)

The factor $(1+\beta^{2})$ in Eq. (2) is discussed in Appendix D together with
other phenomenological corrections.

Figure 1:  Photo-production of a $J/\Psi$ in the dipole model

As indicated in Eq. (1), see also Fig. 1, the $\gamma^{*}\,p$ scattering
proceeds through three steps: (1) First, the incoming virtual photon
fluctuates into a pair of nearly on-shell quark and antiquark; the transverse
size of the pair is ${\bf r}$ and the quark carries a fraction $z$ of the
photon’s light-cone momentum. At high energy the lifetimes of such
fluctuations are much longer than the interaction time, so that the transverse
size and orientation of the color dipole can be considered frozen during the
scattering process Nikolaev and Zakharov (1991, 1992). The splitting of the
photon is described by the virtual photon wave function
$\Psi_{T,L}(r,z,Q^{2})$, which can be calculated perturbatively in QED (see
e.g. Refs. Nemchik  _et al._ (1994, 1997); Kovchegov and Levin (2012)). (2)
The $q\bar{q}$ pair scatters elastically on the proton, with an amplitude
$i{\cal A}_{q\bar{q}}$. The factor $i$ here takes into account the fact that
the amplitude is essentially imaginary (that is, with this factor, ${\cal
A}_{q\bar{q}}$ is real). (3) Finally the scattered dipole recombines to form a
final state, in the present case the vector meson with wave function
$\Psi_{V}(r,z,Q^{2})$.

For simplicity, in this paper we shall focus on photo-production, $Q^{2}=0$
and only the transverse part of the photon wave function contributes. We then
simplify the notation and write the amplitude as

$\displaystyle{\cal A}^{\gamma p\to
Vp^{\prime}}({x_{\\!I\\!\\!P}},\mbox{\boldmath$\Delta$})=i\int d^{2}{\bf
r}\int_{0}^{1}{{\rm d}z\over 4\pi}\,[\Psi_{V}^{*}\Psi_{\gamma}]_{(r,z)}\,{\cal
A}_{q\bar{q}}({x_{\\!I\\!\\!P}},{\bm{r}},\mbox{\boldmath$\Delta$}).$ (3)

The overlap factor $[\Psi_{V}^{*}\Psi_{\gamma}]_{(r,z)}$ of the wave functions
of the photon and the vector meson is discussed in Appendix B. In the next
subsection, we recall briefly the calculation of the amplitude ${\cal
A}_{q\bar{q}}({x_{\\!I\\!\\!P}},{\bm{r}},\mbox{\boldmath$\Delta$})$ in the
dipole model.

###  II.1 The dipole cross section

In this and the next subsection, we consider the scattering of a color dipole
of fixed size on a complex target, such as a proton or a nucleus. We start
with the scattering on a proton, and move to the impact parameter
representation, defining (temporarily, see Eq. (38) below)

${\cal
A}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{r}},\mbox{\boldmath$\Delta$})=\int{\rm
d}^{2}{\bm{b}}\,{\rm e}^{-i{\bm{b}}\cdot\mbox{\boldmath$\Delta$}}{\cal
A}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{r}},{\bm{b}}).$ (4)

At high energy, the scattering amplitude of the dipole is purely imaginary (it
is in the explicit two gluon exchange approximation to be used shortly), so
that the ${\cal S}$-matrix is real and ${\cal S}_{q\bar{q}}=1-{\cal
A}_{q\bar{q}}$. In the eikonal approximation the dipole ${\cal S}$-matrix is
given by the average of the product of two Wilson lines that represent the
propagation of the quark and the antiquark in the color field of the proton:

${\cal
S}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{b}},{\bm{r}})=\frac{1}{N_{c}}\langle{\rm
Tr}\left[U^{\dagger}({\bm{b}}+{\bm{r}}/2)U({\bm{b}}-{\bm{r}}/2)\right]\rangle,$
(5)

the average represented by the angular brackets being made over the color
field of the proton (see e.g. Blaizot (2017)). This average provides the
$x$-dependence of ${\cal S}_{q\bar{q}}(x,{\bm{b}},{\bm{r}})$. A simple
calculation, in the two-gluon exchange approximation, allows us to relate
${\cal S}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{b}},{\bm{r}})$ to the gluon density
$x_{\\!I\\!\\!P}g(x_{\\!I\\!\\!P})$,

${\cal
S}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{b}},{\bm{r}})=1-\frac{1}{2}\sigma_{\rm
dip}(x_{\\!I\\!\\!P},r)T_{G}(b),$ (6)

where $\sigma_{\rm dip}(x_{\\!I\\!\\!P},r)$, the total dipole-proton cross
section, is given by Blaettel  _et al._ (1993a)

$\sigma_{\rm
dip}(x_{\\!I\\!\\!P},r)=\frac{\pi^{2}}{3}r^{2}\alpha_{S}(\mu^{2})\,x_{\\!I\\!\\!P}g(x_{\\!I\\!\\!P},\mu^{2}).$
(7)

The scale $\mu^{2}$ in the gluon density will be specified later (see Eq. (41)
in the next section). $T_{G}({\bm{b}})$ is the average transverse density
profile of the proton, normalized such that $\int{\rm
d}^{2}{\bm{b}}\,T_{G}({\bm{b}})=1$. For a spherical proton, which has been
assumed in writing Eq. (6), the average profile depends only on
$b=|{\bm{b}}|$. Simlarly $\sigma_{\rm dip}$ in Eq. (7) is averaged over the
orientation of the dipole and depends only on $r=|{\bm{r}}|$. In writing Eqs.
(5) and (6), we have assumed that the total gluon density is smoothly
distributed in the transverse plane according to the function $T_{G}(b)$. A
more elaborate treatment will be presented at the end of the next subsection.
At this point, we note that the total cross section is equal to the imaginary
part of the forward $q\bar{q}$ scattering amplitude, that is (omitting to
indicate the $x_{\\!I\\!\\!P}$ dependence to simplify the notation)

$\sigma_{\rm dip}(r)={\cal
A}_{q\bar{q}}({\bm{r}},\mbox{\boldmath$\Delta$}=0)=\int{\rm
d}^{2}{\bm{b}}\,{\cal A}_{q\bar{q}}({\bm{r}},{\bm{b}})=2\int{\rm
d}^{2}{\bm{b}}\left[1-{\mathcal{S}}_{q\bar{q}}({\bm{r}},{\bm{b}})\right],$ (8)

where the last equality involves the standard eikonal expression of the total
cross section in terms of the ${\cal S}$-matrix. This suggests the
identification Munier  _et al._ (2001)

${\cal
A}_{q\bar{q}}({\bm{b}})=2\left[1-{\mathcal{S}}_{q\bar{q}}({\bm{r}},{\bm{b}})\right]\equiv\frac{{\rm
d}\sigma_{q\bar{q}}}{{\rm d}^{2}{\bm{b}}}.$ (9)

The two-gluon exchange calculation is valid in the weak field limit, or
equivalently when the gluon density is small enough (also called dilute
regime). In the dense regime, multiple scattering need to be taken into
account. There are several, more or less equivalent, ways to do that (see e.g.
Blaizot (2017)), the net result being simply the exponentiation of the ${\cal
S}$-matrix, that is

${\cal S}_{q\bar{q}}({\bm{r}},{\bm{b}})=\exp\left(-\frac{1}{2}\sigma_{\rm
dip}(r)T_{G}(b)\right).$ (10)

In this case, the $q\bar{q}$ cross section at a given impact parameter
${\bm{b}}$, Eq. (9), takes the form

$\frac{{\rm d}\sigma_{q\bar{q}}}{{\rm d}^{2}{\bm{b}}}=2\left[1-{\rm
e}^{-\frac{1}{2}\sigma_{\rm dip}(r)T_{G}(b)}\right].$ (11)

This quantity plays the role of the eigenvalue of the scattering matrix, as
will be made more visible shortly. The scattering eigenstates are the states
localized at the transverse positions of the quark and the antiquark, which
positions are frozen during the scattering. All the information about the
proton is contained in the gluon density $x_{\\!I\\!\\!P}g(x_{\\!I\\!\\!P})$
and the average transverse density profile, $T_{G}(b)$. The dependence on the
dipole size is explicit in the dipole cross section, $\sigma_{\rm dip}\propto
r^{2}$. It is then convenient to write ${\cal S}_{q\bar{q}}$ in Eq. (11) as

${\cal S}_{q\bar{q}}(r,b)=\exp\left(-\frac{Q_{s}^{2}(b)r^{2}}{4}\right),$ (12)

where

$Q_{s}^{2}(b)\,r^{2}=2\sigma_{\rm dip}(r)T_{G}(b).$ (13)

Here $Q_{s}(b)$ is the saturation momentum, defined as a function of the
impact parameter. The saturation momentum fixes the scale of what is meant by
“small” or “large” dipole, or equivalently what is, for a fixed dipole size, a
dilute or a dense system. Eq. (12) exhibits the interplay between the
properties of the projectile (e.g. the size of the dipole) and those of the
target (the gluon density that enters the saturation momentum) in the
scattering process (see the discussion at the end of Sect. III.2).

###  II.2 Fluctuations in the target

The average profile function $T_{G}(b)$ introduced above is an average
quantity, related to the local average gluon density in the proton. However
such a quantity may fluctuate event-by event. Such fluctuations are
responsible for the incoherent contribution to the diffractive cross section.
To analyze more clearly this important source of fluctuations, we shall first
consider the scattering of a dipole on a nucleus Caldwell and Kowalski (2010),
leaving for the moment the description of the proton untouched. At the end of
this subsection, we shall discuss analogous fluctuations for the proton,
considered as a composite object made of valence quarks.

Consider a nucleus made of $A$ nucleons. The density profile, that is the
density in the transverse plane (after integration of the three dimensional
nucleon density over the longitudinal coordinate) is given by the convolution
of the nucleon density with the gluon density in the proton, i.e.,

$\hat{T}_{A}({\bm{b}})=\sum_{i=1}^{A}T_{G}({\bm{b}}_{i}-{\bm{b}})=\int{\rm
d}^{2}{\bm{x}}\,\hat{\rho}({\bm{x}})T_{G}({\bm{x}}-{\bm{b}}),$ (14)

where ${\bm{b}}_{i}$ denotes the location of nucleon $i$ and ${\bm{b}}$ is the
impact parameter of the dipole (defined at this point at mid distance between
the quark and the antiquark, see Eq. (5)). $\hat{T}_{A}({\bm{b}})$ is a one-
body operator in the nucleon variables, and $\hat{\rho}({\bm{x}})$ denotes the
nucleon density operator in the transverse plane:

$\hat{\rho}({\bm{x}})=\sum_{i=1}^{A}\delta({\bm{x}}-{\bm{b}}_{i}).$ (15)

The matrix element
$\langle\Psi_{0}|\hat{T}_{A}({\bm{b}})|\Psi_{0}\rangle=\langle\hat{T}_{A}({\bm{b}})\rangle$
is easily evaluated in terms of the average density
$\langle\hat{\rho}({\bm{x}})\rangle=\langle\Psi_{0}|\hat{\rho}({\bm{x}})|\Psi_{0}\rangle$,
$\langle\hat{T}_{A}({\bm{b}})\rangle=\int{\rm
d}^{2}{\bm{x}}\,\langle\hat{\rho}({\bm{x}})\rangle T_{G}({\bm{x}}-{\bm{b}}).$
Note that $\int{\rm d}^{2}{\bm{b}}\,\hat{T}_{A}({\bm{b}})=A$.

The expressions (6) and (9) giving the cross section at a given impact
parameter in the weak field limit generalize into

$\frac{{\rm d}\sigma_{q\bar{q}}}{{\rm d}^{2}{\bm{b}}}=\sigma_{\rm
dip}(r)\hat{T}_{A}({\bm{b}}),$ (16)

which is now an operator in the nucleon variables. This operator is diagonal
in the basis of states $|{\bm{b}}_{1},\cdots,{\bm{b}}_{A}\rangle$, where the
${\bm{b}}_{i}$’s are the positions of the individual nucleons, frozen during
the collision process: these states can be considered as diffraction
eigenstates Miettinen and Pumplin (1978). The elastic amplitude is obtained by
taking the average of this cross section in the ground state wave function,
which amounts to average over the distribution of the positions
${\bm{b}}_{i}$. At a given momentum transfer $\Delta$, the coherent
diffractive cross section of a dipole on a nucleus is then proportional to

$\int{\rm d}^{2}{\bm{b}}\,{\rm d}^{2}{\bm{b}}^{\prime}{\rm
e}^{-i\mbox{\boldmath$\Delta$}\cdot({\bm{b}}-{\bm{b}}^{\prime})}\left\langle\frac{{\rm
d}\sigma_{q\bar{q}}}{{\rm d}^{2}{\bm{b}}}\right\rangle\left\langle\frac{{\rm
d}\sigma_{q\bar{q}}}{{\rm
d}^{2}{\bm{b}}^{\prime}}\right\rangle=\langle\Sigma_{q\bar{q}}(\mbox{\boldmath$\Delta$})\rangle^{2},$
(17)

where

$\Sigma_{q\bar{q}}(\Delta)\equiv\int{\rm d}^{2}{\bm{b}}\,{\rm
e}^{-i\Delta\cdot{\bm{b}}}\,\frac{{\rm d}\sigma_{q\bar{q}}}{{\rm
d}^{2}{\bm{b}}}$ (18)

and the angular brakets denote average over the ground state wave function.

In the dense regime we have

$\frac{{\rm d}\sigma_{q\bar{q}}}{{\rm d}^{2}{\bm{b}}}=2\left(1-{\rm
e}^{-\frac{1}{2}\sigma_{\rm dip}\hat{T}_{A}({\bm{b}})}\right).$ (19)

This is no longer a one-body operator, and its average can no longer be
expressed in terms of the average of $\hat{T}_{A}({\bm{b}})$. However, it is
straightforward to sample the probability distribution associated with the
ground state wave function, viz.
$|\langle{\bm{b}}_{1},\cdots,{\bm{b}}_{A}|\Psi_{0}\rangle|^{2}$, and proceed
to a numerical evaluation of the average of the cross section (19) (see next
section).

Similar considerations apply to the total diffractive cross section. The
elastic amplitude assumes that only the ground state of the nucleus
contributes as an intermediate state. We can relax this assumption and allow
all possible diffraction eigenstates as intermediate states. Thus, in the weak
field limit, we are led to define the total diffractive cross section as

$\int{\rm d}^{2}{\bm{b}}\,{\rm d}^{2}{\bm{b}}^{\prime}{\rm
e}^{-i\mbox{\boldmath$\Delta$}\cdot({\bm{b}}-{\bm{b}}^{\prime})}\,\sigma_{\rm
dip}^{2}\sum_{\alpha}\left|\langle\alpha|\hat{T}_{A}({\bm{b}})|\Psi_{0}\rangle\right|^{2}=\langle\Sigma^{2}_{q\bar{q}}(\mbox{\boldmath$\Delta$})\rangle,$
(20)

where $|\alpha\rangle=|{\bm{b}}_{1},\cdots,{\bm{b}}_{A}\rangle$ denotes a
scattering eigenstate. By combining with Eq. (17), one finds that the
incoherent cross section (in the weak field limit) is proportional to

$\langle\Sigma^{2}_{q\bar{q}}(\mbox{\boldmath$\Delta$})\rangle-\langle\Sigma_{q\bar{q}}(\mbox{\boldmath$\Delta$})\rangle^{2}.$
(21)

That is, the cross section is proportional to the fluctuation of the dipole
cross section, a familiar result of the Good-Walker formalism Good and Walker
(1960); Miettinen and Pumplin (1978).

The evaluation of
$\sum_{\alpha}\left|\langle\alpha|\hat{T}_{A}({\bm{b}})|\Psi_{0}\rangle\right|^{2}$
can conveniently be done in terms of the density-density correlation function.
Considering $|\langle{\bm{b}}_{1},\cdots,{\bm{b}}_{A}|\Psi_{0}\rangle|^{2}$ as
a symmetric probability distribution, we define the associated one and two-
point functions

$\int{\rm d}^{2}{\bm{b}}_{1}\cdots{\rm
d}^{2}{\bm{b}}_{A}|\langle{\Psi_{0}}|{\bm{b}}_{1},{\bm{b}}_{2},\cdots,{\bm{b}}_{A}\rangle|^{2}\,\sum_{i=1}^{A}\delta({\bm{x}}-{\bm{b}}_{i})=A\rho^{(1)}({\bm{x}}),$
(22)

and

$\displaystyle\int{\rm d}^{2}{\bm{b}}_{1}\cdots{\rm
d}^{2}{\bm{b}}_{A}|\langle{\Psi_{0}}|{\bm{b}}_{1},{\bm{b}}_{2},\cdots,{\bm{b}}_{A}\rangle|^{2}\,\sum_{i,j}\delta({\bm{x}}-{\bm{b}}_{i})\delta({\bm{y}}-{\bm{b}}_{j})$
$\displaystyle\qquad\qquad=A\rho^{(1)}({\bm{x}})\delta({\bm{x}}-{\bm{y}})+A(A-1)\rho^{(2)}({\bm{x}},{\bm{y}}).$
(23)

These are normalized to unity: $\int{\rm
d}^{2}{\bm{x}}\,\rho^{(1)}({\bm{x}})=1$, $\int{\rm d}^{2}{\bm{x}}\,{\rm
d}^{2}{\bm{y}}\,\rho^{(2)}({\bm{x}},{\bm{y}})=1$. Note also that
$\langle\hat{\rho}({\bm{x}})\rangle=A\rho^{(1)}({\bm{x}})$. The density-
density correlation function reads111We use a standard notation,
$S({\bm{x}},{\bm{y}})$, for the correlation function (see e.g. Blaizot  _et
al._ (2014) for a discussion of this object in the context of Glauber Monte
Carlo simulations). This should not be confused with the ${\cal S}$-matrix.

$\displaystyle
S({\bm{x}},{\bm{y}})=\langle\hat{\rho}({\bm{x}})\hat{\rho}({\bm{y}})\rangle-\langle\hat{\rho}({\bm{x}})\rangle\langle\hat{\rho}({\bm{y}})\rangle=A\rho^{(1)}({\bm{x}})\delta({\bm{x}}-{\bm{y}})\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+A(A-1)\rho^{(2)}({\bm{x}},{\bm{y}})-A\rho^{(1)}({\bm{x}})A\rho^{(1)}({\bm{y}}).$
(24)

In the absence of correlations between the particles,
$\rho^{(2)}({\bm{x}},{\bm{y}})=\rho^{(1)}({\bm{x}})\rho^{(1)}({\bm{y}})$ and

$S({\bm{x}},y)=A\rho^{(1)}({\bm{x}})\delta({\bm{x}}-{\bm{y}})-A\rho^{(1)}({\bm{x}})\rho^{(1)}({\bm{y}}).$
(25)

The first term represents the local fluctuation of the density. This is the
source of the fluctuations responsible for the incoherent scattering. The last
term in the formula above just ensures that $\int{\rm
d}^{2}{\bm{x}}\,\,S({\bm{x}},{\bm{y}})=0$, an equality which holds when the
total number of nucleons does not fluctuate.

We now return to the evaluation of the fluctuation of the nucleus profile
$\hat{T}_{A}({\bm{b}})$, and note that

$\displaystyle\langle\hat{T}_{A}({\bm{b}})\hat{T}_{A}({\bm{b}}^{\prime})\rangle-\langle\hat{T}_{A}({\bm{b}})\rangle\langle\hat{T}_{A}({\bm{b}}^{\prime})\rangle=\int{\rm
d}^{2}{\bm{x}}\int{\rm
d}^{2}{\bm{y}}\,T_{G}({\bm{x}}-{\bm{b}})T_{G}({\bm{y}}-{\bm{b}}^{\prime})S({\bm{x}},{\bm{y}})$
$\displaystyle\qquad\qquad\simeq A\int{\rm
d}^{2}{\bm{x}}\rho^{(1)}({\bm{x}})T_{G}({\bm{x}}-{\bm{b}})T_{G}({\bm{x}}-{\bm{b}}^{\prime})-\frac{1}{A}\langle\hat{T}_{A}({\bm{b}})\rangle\langle\hat{T}_{A}({\bm{b}}^{\prime})\rangle,$
(26)

where, in the second line, we have assumed uncorrelated nucleons (hence the
approximate equal sign). In Fourier space, Eq. (II.2) yields

$\int{\rm d}^{2}{\bm{b}}\,{\rm d}^{2}{\bm{b}}^{\prime}{\rm
e}^{-i\mbox{\boldmath$\Delta$}\cdot({\bm{b}}-{\bm{b}}^{\prime})}\left[\int{\rm
d}^{2}{\bm{x}}\,\rho({\bm{x}})T_{G}({\bm{x}}-{\bm{b}})T_{G}({\bm{x}}-{\bm{b}}^{\prime})-\frac{1}{A}\langle
T_{A}({\bm{b}})\rangle\langle T_{A}({\bm{b}}^{\prime})\right].$ (27)

At small $\Delta$ the integration over the impact parameters becomes free, and
the integral goes to zero. This implies that the fluctuations related to the
random locations of the sources of the gluon field (here the nucleons) vanish.
This simple relation of the fluctuation of the cross section to the density-
density correlation function holds strictly in the dilute regime, where the
cross section is given by Eq. (16), but should provide a reasonable
approximation even in the dense regime222Note for instance that, for the
relevant values of parameters, the expansion in $\sigma_{\rm dip}$ appears to
converge rapidly, as indicated in Appendix A.. It explains the suppression of
the geometrical fluctuations observed in the incoherent cross section at small
momentum transfer (see the discussion at the end of Sect. III.1).

The previous discussion concerned the scattering on a nucleus. One can extend
it to the case of a proton, assimilating the valence quarks to the sources of
the gluon field Mäntysaari and Schenke (2016b). All we need to do is to
replace the average density profile of the proton, well given
phenomenologically by a Gaussian,

$T_{G}({\bf b})={1\over 2\pi B_{G}}e^{-{{\bf b}^{2}/(2B_{G})}},$ (28)

by the contribution of $N_{q}=3$ valence quarks

$T_{G}({\bf b})\mapsto\hat{T}_{G}({\bm{b}})={1\over
N_{q}}\sum_{i=1}^{N_{q}}T_{q}({\bf b}-{\bf b}_{i}).$ (29)

Here

$T_{q}({\bf b})={1\over 2\pi B_{q}}e^{-{\bf b}^{2}/(2B_{q})}$ (30)

plays the role of $T_{G}$ earlier: it represents the average gluon density
around a valence quark, with the parameter $B_{q}$ controlling the size of
this gluon cloud. In short, in this picture, valence quarks play the role of
the nucleons, and the proton is considered as a bound state of valence quarks.

###  II.3 Fluctuations in the projectile

Aside from the geometrical fluctuations in the target that we have just
discussed, one can also expect fluctuations in the projectile arising from the
quantum mechanical splitting of the photon into a quark-antiquark pair: the
resulting color dipole can have, event-be-event, a different size and
orientation. The amplitude (3) is explicitly written as an average over these
dipole degrees of freedom, the overlap of the photon and the vector meson wave
functions playing the role of a distribution of dipole sizes Munier  _et al._
(2001). However this overlap is not quite the square of a normalized wave
function, and the “non-diagonal” character of this object brings in subtleties
that need to be dealt with. We shall keep the discussion general, call
$\psi_{i}$ the initial wave function (that of the photon) and $\psi_{f}$ the
final one (that of the vector meson), and ignore the geometrical fluctuations
of the proton. By expanding $\psi_{i}$ and $\psi_{f}$ on the diffraction
eigenstates (here color dipoles of various sizes), we get

$|\psi_{i}\rangle=\sum_{\alpha}C_{i}^{\alpha}|\alpha\rangle,\quad|\psi_{f}\rangle=\sum_{\alpha}C_{f}^{\alpha}|\alpha\rangle,\quad\langle\psi_{f}|\psi_{i}\rangle=\sum_{\alpha}C^{\alpha*}_{f}C_{i}^{\alpha}.$
(31)

The elastic (or coherent) scattering amplitude is then given by

$\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle=\sum_{\alpha}C_{f}^{\alpha*}C_{i}^{\alpha}\,t_{\alpha},\qquad{\cal
T}|\alpha\rangle=t_{\alpha}|\alpha\rangle.$ (32)

Here ${\cal T}$ stands for the (imaginary) scattering matrix and the states
$\alpha$ are its eigenstates, the corresponding eigenvalues being denoted
$t_{\alpha}$. To interpret Eq. (32) as an average, we divide
$C^{f*}_{\alpha}C^{i}_{\alpha}$ by the overlap. We obtain

$\frac{\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=\sum_{\alpha}\frac{C_{f}^{\alpha*}C_{i}^{\alpha}}{\langle\psi_{f}|\psi_{i}\rangle}t_{\alpha}=\sum_{\alpha}p_{\alpha}t_{\alpha}=\langle
t\rangle,$ (33)

where the probabilities $p_{\alpha}$ are properly normalised,
$\sum_{\alpha}p_{\alpha}=1$. In terms of the ${\cal S}$-matrix, this relation
takes the form

$\langle\psi_{f}|{\cal
S}|\psi_{i}\rangle=\langle\psi_{f}|\psi_{i}\rangle-\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle=\langle\psi_{f}|\psi_{i}\rangle\left[1-\frac{\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}\right].$ (34)

The overlap factor gives the contribution that exists in the absence of
interaction. The second term within the squared brackets correctly measures
the effect of the interactions relative to the unit factor.

Similarly, the total diffractive cross section is written in the following way

$\frac{\sum_{\alpha}\langle\psi_{f}|{\cal T}|\alpha\rangle\langle\alpha|{\cal
T}^{\dagger}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=\sum_{\alpha}\frac{C_{f}^{\alpha*}C_{i}^{\alpha}}{\langle\psi_{f}|\psi_{i}\rangle}t_{\alpha}^{2}=\langle
t^{2}\rangle.$ (35)

Leaving aside the Fourier transform, this expression is analogous to the sum
over the diffraction eigenstates in Eq. (20). However, aside from the
normalization associated to the overlap, an important difference lies in the
fact that in the target case, $\psi_{i}=\psi_{f}=\Psi_{0}$, with $\Psi_{0}$
the proton ground state. While $\psi_{i}$ and $\psi_{f}$ do refer to states
with the same quantum numbers, Eq. (35) represents a non trivial
extrapolation.

If one accepts Eq. (35), we are led to expect the incoherent diffractive
cross-section to be proportional to

$\frac{\langle\psi_{f}|{\cal T}{\cal
T}^{\dagger}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}-\left|\frac{\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}\right|^{2}=\langle
t^{2}\rangle-\langle t\rangle^{2},$ (36)

where $\langle\psi_{f}|{\cal T}{\cal T}^{\dagger}|\psi_{i}\rangle$ stands for
$\sum_{\alpha}\langle\psi_{f}|{\cal T}|\alpha\rangle\langle\psi_{i}|{\cal
T}|\alpha\rangle^{*}$. This equation has the correct property to vanish when
$t_{\alpha}$ does not depend on $\alpha$. Indeed, when all the states $\alpha$
are absorbed at the same rate there can be no diffractive effects.

Now, in the calculation of the cross section, we need to reinstate the overlap
in such a way that the coherent cross section is properly normalized. This
implies that one should multiply the result in Eq. (36) by the overlap
squared. Thus, to within overall numerical factors independent of $\psi_{i}$
and $\psi_{f}$, the incoherent diffractive cross section ends up being
proportional to

$\langle\psi_{f}|{\cal T}{\cal
T}^{\dagger}|\psi_{i}\rangle\langle\psi_{f}|\psi_{i}\rangle-\left|\langle\psi_{f}|{\cal
T}|\psi_{i}\rangle\right|^{2}=\langle\psi_{f}|\psi_{i}\rangle^{2}\left[\langle
t^{2}\rangle-\langle t\rangle^{2}\right].$ (37)

Note that when $\psi_{i}=\psi_{f}=\Psi_{0}$ one recovers the standard result
of the previous subsection.

##  III  Event-by-event calculation

We turn now to the explicit calculation of the photoproduction cross section
of vector mesons in the dipole model, Eq. (2), based on the explicit
expression of the amplitude given in Eq. (1). We specialize to the case of
$J/\Psi$ production, and consider both coherent and incoherent contributions
to the cross section.

Let us start with a remark concerning the Fourier transform of the amplitude.
In the previous section this was given by Eq. (4). In fact there is a subtlety
here, related to the definition of the impact parameter. It may be chosen as
the transverse distance between the center of mass of the proton and that of
the dipole, the latter coinciding with the location of the photon in the
transverse plane. Now, the center of mass of the dipole depends on the
fraction $z$ of the longitudinal momentum carried by one member of the dipole.
If ${\bm{x}}_{1}$ and ${\bm{x}}_{2}$ denote respectively the transverse
coordinates of the quark and that of the antiquark, as measured from the
center of mass of the proton, then ${\bm{r}}={\bm{x}}_{1}-{\bm{x}}_{2}$ and
${\bm{b}}=z{\bm{x}}_{1}+(1-z){\bm{x}}_{2}$. The ${\cal S}$-matrix element to
be evaluated is therefore of the form ${\cal
S}({\bm{b}}+(1-z){\bm{r}},{\bm{b}}-z{\bm{r}})$ which differs from the
expression (5) in which ${\bm{b}}$ is measured from the midpoint between the
quark and the antiquark. However, we can write ${\cal
S}({\bm{b}}+(1-z){\bm{r}},{\bm{b}}-z{\bm{r}}))={\cal
S}({\bm{b}}^{\prime}+{\bm{r}}/2,{\bm{b}}^{\prime}-{\bm{r}}/2)$ with
${\bm{b}}^{\prime}={\bm{b}}+\left(\frac{1}{2}-z\right){\bm{r}}$, and a simple
change of variables allows us to write the amplitude (4) in the form 333The
factor $1/2-z$ in Eq. (38) differs from the factor $1-z$ often used in that
context (see e.g. Kowalski  _et al._ (2006); Mäntysaari and Schenke (2016b),
including our previous work Traini and Blaizot (2019)). The present discussion
is inspired from Ref. Hatta  _et al._ (2017).

$\Sigma_{q\bar{q}}({x_{\\!I\\!\\!P},{\bf r},z,\mbox{\boldmath$\Delta$}})=\int
d^{2}{\bf b}{{d\sigma_{q\bar{q}}\over d^{2}{\bf b}}}({\bf
b},{r},{x_{\\!I\\!\\!P}})\,e^{-i({\bf b}-[(1/2-z){\bf
r}])\cdot{\mbox{\boldmath$\Delta$}}},$ (38)

where the variable conjugate to $\Delta$ is not simply ${\bm{b}}$ as in (4),
but ${\bm{b}}-(\frac{1}{2}-z){\bm{r}}$. Note that
$\Sigma_{q\bar{q}}({x_{\\!I\\!\\!P},{\bf r},z,\mbox{\boldmath$\Delta$}})$
differs from the amplitude ${\cal
A}_{q\bar{q}}(x_{\\!I\\!\\!P},{\bm{r}},\mbox{\boldmath$\Delta$})$ solely by
the $z$-dependent phase factor.

The incoherent photo-production cross section can be written in the form

$\displaystyle{16\pi\over(1+\beta^{2})}\,{d\sigma^{\gamma p\to
Vp^{\prime}}\over dt}=\bigg{<}\left|{\cal A}^{\gamma p\to
Vp^{\prime}}({x_{\\!I\\!\\!P}},\mbox{\boldmath$\Delta$})\right|^{2}\bigg{>}-\left|\bigg{<}{\cal
A}^{\gamma p\to
Vp^{\prime}}({x_{\\!I\\!\\!P}},\mbox{\boldmath$\Delta$})\bigg{>}\right|^{2}$

where the second term give the elastic, or coherent, cross section (see Eq.
(2)). The angular brackets represent the various averages over the projectile
and the target. These averages will be calculated by sampling appropriate
distributions, as will be explained in the following subsections.

###  III.1 Proton shape fluctuations

In this subsection, we review the effects of proton shape fluctuations,
ignoring the fluctuations of the dipole sizes. Following Mäntysaari and
Schenke (2016a) we assume that gluons are distributed around the valence
quarks, geometrical fluctuations arising from the random locations of the
valence quarks, event-by-event. To deal with these, we replace the average
density profile of the proton, $T_{G}(b)$, by the expression (29) involving
the sum of gluon densities attached to the valence quarks.444Recently Traini
and Blaizot (2019) an approach connecting the density profile to the valence
quark distribution in the proton has been proposed. The approach does not need
any free parameters even for the gluon distribution which is directly
connected to the valence quark density by means of appropriate DGLAP
evolution. Typical parameters for the various Gaussian distributions involved
in Eqs. (28, 29, 30) are respectively $B_{G}=4\;{\rm GeV}^{-2}$,
$B_{qc}=3.3\;{\rm GeV}^{-2}$, and $B_{q}=0.7\;{\rm GeV}^{-2}$ (note that
$B_{G}=B_{qc}+B_{q}$). In terms of root mean square radii $\sqrt{2B}$, these
numbers correspond to $0.56\;{\rm fm},0.51\;{\rm fm}$ and $0.24\;{\rm fm}$
respectively for the radius of the proton, that of the valence quark
distribution and the size of the gluon cloud around each valence quark. A
sampling procedure selects randomly a large number of quark positions ${\bf
b}^{\\{c\\}}_{i}$ in a Gaussian distribution with parameter $B_{qc}$. A given
choice of ${\bf b}^{\\{c\\}}_{i}$ is referred to as a configuration. The cross
sections are then obtained by adding the contributions of each configuration,
and dividing by the total number $N_{C}$ of these configurations (see Eq.
(III.1) below). The results presented below were obtained for $N_{C}=10\,000$
configurations, as in ref. Traini and Blaizot (2019).

The dipole cross section depends on the configuration, and we write

$\displaystyle{d\sigma_{q\bar{q}}^{\\{c\\}}\over d^{2}{\bf
b}}(x_{\\!I\\!\\!P},{\bf b},{\bf r})=2\left[1-\exp\bigg{(}-{1\over
2}\sigma_{\rm dip}(x_{\\!I\\!\\!P},{\bm{r}})\,T^{\\{c\\}}_{G}({\bf
b})\bigg{)}\right],$ (40)

where $T^{\\{c\\}}_{G}({\bf b})$ is evaluated according to Eq. (29) in which
${\bm{b}}_{i}\mapsto{\bm{b}}_{i}^{\\{c\\}}$. The Fourier transform of this
expression, according to Eq. (38), yields
$\Sigma^{\\{c\\}}_{q\bar{q}}({x_{\\!I\\!\\!P},{\bf
r},z,\mbox{\boldmath$\Delta$}})$. The integration over the impact parameter
involved in the Fourier transform is done as explained in Appendix A. The
scale $\mu$ in the gluon distribution function
${x_{\\!I\\!\\!P}}g({x_{\\!I\\!\\!P}},\mu^{2})$ that enters $\sigma_{\rm
dip}(x_{\\!I\\!\\!P},{\bm{r}})$ (see Eq. (7)) is related to the size ${r}$ of
the dipole Bartels  _et al._ (2002)

$\mu^{2}=\mu^{2}({r}^{2})=\mu_{0}^{2}+{4\over{r}^{2}},$ (41)

and the gluon distribution is parameterized as

$x_{\\!I\\!\\!P}g(x_{\\!I\\!\\!P},\mu_{0}^{2})=A_{g}\,x_{\\!I\\!\\!P}^{-\lambda_{g}}\,(1-x_{\\!I\\!\\!P})^{5.6}.$
(42)

We use the parameters given in Ref. Bartels  _et al._ (2002).

Since, in this subsection, we are ignoring the fluctuations in the dipole
degrees of freedom, we define an average $\bar{\Sigma}$ by integrating over
${\bm{r}}$ and $z$,

$\bar{\Sigma}^{\\{c\\}}_{q\bar{q}}(\mbox{\boldmath$\Delta$})=\int d^{2}{\bf
r}\int{dz\over 4\pi}\left[\Psi^{*}_{\gamma}\Psi_{V}\right]_{({\bf
r},z)}\Sigma^{\\{c\\}}_{q\bar{q}}({{\bf r},z,\mbox{\boldmath$\Delta$}}).$ (43)

Physically $\bar{\Sigma}$ represents the cross section of an average dipole
scattering on a fluctuating proton. In terms of $\bar{\Sigma}$, the incoherent
cross section (III) takes then the simple form

$\displaystyle{16\pi\over(1+\beta^{2})}\,{d\sigma^{\gamma p\to
Vp^{\prime}}\over dt}={1\over
N_{C}}\sum_{\\{c\\}}\bigg{|}\bar{\Sigma}^{\\{c\\}}_{q\bar{q}}({\bf
r},z,\mbox{\boldmath$\Delta$})\bigg{|}^{2}-\left|{1\over
N_{C}}\sum_{\\{c\\}}\bar{\Sigma}^{\\{c\\}}_{q\bar{q}}({{\bf
r},z,\mbox{\boldmath$\Delta$}})\right|^{2},$

where the sums run over the configurations defined above. The coherent cross
section is given by the second term in this equation.

Figure 2: (color on line) Coherent and incoherent cross sections for $J/\Psi$
photoproduction at $W=75$ GeV. The dashed and dotted lines represent
respectively the coherent and incoherent cross sections calculated according
to Eq. (III.1). Coherent H1 data from Refs. H1 collaboration  _et al._ (2006,
2013) (circles), incoherent data (triangles) from H1 and ZEUS experiments of
Refs. H1 collaboration  _et al._ (2013); ZEUS collaboration  _et al._ (2003).

Fig. 2 shows the results of calculations of both coherent and incoherent cross
sections for HERA kinematics and $J/\Psi$ photo-production. Only the
fluctuations of the proton shape are taken into account in the calculation.

At small momentum transfer $\Delta$, the cross section is dominated by the
elastic component. This decreases exponentially, $\propto{\rm e}^{-|t|B_{D}}$
where the parameter $B_{D}$ is of the order of $B_{G}$ (but slightly larger
Caldwell and Kowalski (2010)). That is, the exponential drop of the coherent
cross section is essentially determined by the size of the gluon cloud in the
proton. This is clear from Eq. (16) showing that in the weak field limit, the
average $\langle\Sigma(\mbox{\boldmath$\Delta$})\rangle$ is proportional to
the Fourier transform of the average $\langle\hat{T}_{G}({\bm{b}})\rangle$.

The coherent contribution drops rapidly, and ceases to be dominant when the
geometrical fluctuations start to become important, which occurs when
$|t|\gtrsim 0.4-0.5$ Gev2, corresponding to a typical length scale of order
0.3 fm. The incoherent cross section decreases with a much smaller slope than
the coherent cross section, the latter becoming rapidly negligible as $\Delta$
increases. The good agreement with the data confirms that geometric
fluctuations associated to a “lumpy” proton appears to be a crucial ingredient
to reproduce the incoherent cross section Mäntysaari and Schenke (2016a) at
moderate momentum transfer, $|t|\lesssim 2.5$ GeV2.

The same Fig. 2 reveals that, at very low momentum transfer, the effect of
geometrical fluctuations tends to vanish. This feature, already visible in the
early model of Ref. Miettinen and Pumplin (1978), is in fact common to most
hot spot models. It has a simple origin. The gluon density “seen” by a dipole
at a given impact parameter ${\bf b}$, fluctuates event-by-event depending on
whether there are other valence quarks in the vicinity of ${\bf b}$. However,
at small momentum transfer, ${\bf b}$ is averaged over areas of the order or
larger than the proton size. As a result, the dipole “sees” an average proton
density profile, in which density fluctuations are averaged out. A more formal
argument relies on the properties of the density-density correlation function
that we have recalled in Sect. II.2 (see Eq. (27)). Note that if one allows
the total number of nucleons $A$ to fluctuate, one gets a contribution at
$\mbox{\boldmath$\Delta$}=0$. Indeed, in this case,

$\int{\rm d}^{2}{\bm{b}}\,{\rm
d}^{2}{\bm{b}}^{\prime}\,\left[\langle\hat{T}_{A}({\bm{b}})\hat{T}_{A}({\bm{b}}^{\prime})\rangle-\langle\hat{T}_{A}({\bm{b}})\rangle\langle\hat{T}_{A}({\bm{b}}^{\prime})\rangle\right]=\langle
A^{2}\rangle-\langle A\rangle^{2}.$ (45)

This observation may be exploited in hot spot models to compensate for the
vanishing of the incoherent cross section at vanishing $\Delta$. However, this
is not needed since, as we show in the next section, precisely that region of
small momentum transfer is dominated by fluctuations of the dipole size in the
initial state.

###  III.2 Dipole size fluctuations

In order to calculate the fluctuation of the dipole size, we assimilate the
overlap between the photon and the $J/\Psi$ meson to a probability
distribution, as we have already mentioned. More precisely, we set

${\cal P}(r)\propto{2\pi r}\int{dz\over
4\pi}\,[\Psi^{*}_{V}\Psi_{\gamma}]_{(r,z)},\qquad\int dr{\cal P}(r)=1.$ (46)

That is, we define the probability ${\cal P}(r)$ after integration over $z$.
Since we shall keep the $z$-average intact, this introduces a slight
inconsistency in the calculation which is of minor relevance since the effect
of the $z$-dependence is in any case small. The distribution ${\cal P}(r)$ is
plotted in Fig. 3. This distribution is sampled in order to get an event-by-
event realization of random dipole sizes. In our calculation, we use
$N_{C}=10\,000$ configurations chosen with the sampling procedure detailed in
Appendix C.

At this point, in order to make the calculation clearer, we ignore temporarily
the proton shape fluctuations, that is, $\Sigma$ in Eq.
(LABEL:eq:dxsection_incoh_fluct2) below is averaged over the proton ground
state. In fact we shall use the simple Gaussian approximation in which
$\Sigma$ is calculated with the average profile $T_{G}(b)$. We shall denote
$\tilde{\Sigma}$ the corresponding average of $\Sigma$. That is
$\tilde{\Sigma}$ is calculated by substituting in Eq. (38) the expression (11)
for ${\rm d}\sigma/{\rm d}^{2}{\bm{b}}$.

Figure 3: Photon - $J/\Psi$ transverse overlap function (46) integrated over
$z$.

Each event (which we shall also refer to as a configuration) corresponds to a
choice of a dipole size $r_{\\{c\\}}=|{\bf r}_{\\{c\\}}|$ extracted from the
probability density (46) with the appropriate frequency (see Fig. 3). The
average of the amplitude over the dipole degrees of freedom (${\bm{r}}$ and
$z$) can therefore be written as

$\displaystyle\bigg{<}{\cal A}^{\gamma p\to
Vp^{\prime}}(\mbox{\boldmath$\Delta$})\bigg{>}$ $\displaystyle=$
$\displaystyle\int d^{2}{\bf r}\int{dz\over
4\pi}\left[\Psi^{*}_{\gamma}\Psi_{V}\right]_{({\bf
r},z)}\tilde{\Sigma}_{q\bar{q}}({{\bf r},z,\mbox{\boldmath$\Delta$}})$
$\displaystyle\simeq$ $\displaystyle{1\over N_{C}}\sum_{\\{c\\}}\int d^{2}{\bf
r}\int{dz\over 4\pi}\left[\Psi^{*}_{\gamma}\Psi_{V}\right]_{({\bf
r},z)}\tilde{\Sigma}_{q\bar{q}}({{\bf
r}_{\\{c\\}},z,\mbox{\boldmath$\Delta$}}).$

The second line makes explicit that neither the average over $z$ nor the
angular average over the orientation of the dipole are part of the
“configurations” which, in the present case just represent the dipole sizes.
The average over the dipole size is performed via the sum over the
configurations, that is over a selected sample of dipole sizes. The
approximate equal sign in the second line emphasizes that this calculation is
not quite exact since, as indicated earlier, the sampling is done with a
probality distribution that is integrated over $z$ rather than with a $z$
dependent distribution. The $r$-integration in the second line of the equation
above does not affect $\Sigma_{q\bar{q}}({{\bf
r}_{\\{c\\}},z,\mbox{\boldmath$\Delta$}})$. Although this may seem confusing
at first sight, it is convenient to keep the writing as it is because it
exhibits the proper normalization of the amplitude, as explained in Sect.
II.3.

A similar procedure is followed for the total diffractive cross section, which
we may write as follows

$\displaystyle\bigg{<}\left|{\cal A}^{\gamma p\to
Vp^{\prime}}(\mbox{\boldmath$\Delta$})\right|^{2}\bigg{>}\simeq{1\over
N_{C}}\sum_{\\{c\\}}\bigg{|}\int d^{2}{\bf r}\int{dz\over
4\pi}\left[\Psi^{*}_{\gamma}\Psi_{V}\right]_{({\bf
r},z)}\tilde{\Sigma}_{q\bar{q}}({\bf
r}_{\\{c\\}},z,\mbox{\boldmath$\Delta$})\bigg{|}^{2}.$ (48)

As before the role of the $r$-integration is to ensure the proper
normalization. It does not affect $\tilde{\Sigma}({\bf r}_{\\{c\\}})$, which,
we recall, is averaged over the ground state density of the proton.

Figure 4: (color on line) Incoherent $J/\Psi$ photo-production cross section
at the HERA (H1) kinematics $W=75$ GeV. The dashed and dotted lines represent
the incoherent cross sections calculated respectively from Eqs.
(LABEL:eq:dxsection_incoh_fluct2) and (48) for the dipole size fluctuations,
and Eq. (III.1) for the proton shape fluctuations. The full line represents
the total incoherent cross section, estimated here as the sum of the two (we
assume that the two types of fluctuations are only weakly correlated). Data as
in Fig. 2.

The contribution of dipole size fluctuations to the incoherent diffractive
cross section in then obtained from the general expression (III), in which the
needed averages are given in Eqs. (LABEL:eq:dxsection_incoh_fluct2) and (48).
An explicit numerical calculation based on these equations is shown in Fig. 4.
The result is compared to the contribution of the geometrical shape
fluctuations as estimated in the previous subsection, as well as to the data
on $J/\Psi$ photo-production obtained at HERA. The effects of size
fluctuations are substantial at low momentum transfer while for $|t|\gtrsim
0.5$ Gev2 their contribution is hidden under that of the gluon shape
fluctuations. As the plot reveals, the inclusion of the dipole size
fluctuations allows us to reproduce the data in the region of small momentum
transfer. Given that the calculation contains no additional parameter, this
agreement is gratifying.

Dipole size fluctuations induce fluctuations in the dipole cross section. This
makes the present discussion reminiscent of the fluctuating cross sections
introduced in Blaettel  _et al._ (1993a, b). It is also clear, from Eq. (12)
for instance, that fluctuations in the dipole size can be viewed as
fluctuations of the saturation momentum of the proton. In fact, in Ref.
Mäntysaari and Schenke (2016a) $Q_{s}$-fluctuations have been considered
within the IP-Glasma framework, guided by the interpretation of multiplicity
distributions and rapidity correlations in $p+p$ collisions in terms of
saturation scale fluctuations McLerran and Tribedy (2016); Bzdak and Dusling
(2016). After some adjustment of the $Q_{s}$-distribution, a reasonable
agreement with the data was achieved Mäntysaari and Schenke (2016a). As
mentioned above, varying the number of hot spots in hot spot models can also
generate a contribution at small momentum transfer. We believe however that
the dipole size fluctuations discussed in the present paper provide a more
natural explanation, and can be estimated without any additional parameters or
adjustments. Together with the transverse proton shape fluctuations, it
provides a consistent picture of the incoherent cross section for moderate
momentum transfers.

##  IV Summary and conclusions

In this paper we have considered the photo-production of $J/\Psi$ mesons
within the dipole model. We have extended the approach of Ref. Mäntysaari and
Schenke (2016a); Traini and Blaizot (2019) dealing with event-by-event
fluctuations of the proton shape to include the fluctuations of the $q\bar{q}$
dipole size. We have shown that the latter are significant at small momentum
transfer, where the impact parameter of the dipole is integrated almost freely
over the size of the proton. The dipole then sees essentially an average
proton, in which the shape fluctuations of the proton are averaged out (as
follows from a simple argument based on the analysis of the density-density
correlation function). We have suggested a way to treat the fluctuations of
the size of the dipole in which the splitting of the incoming virtual photon
into a quark - antiquark pair is a quantum mechanical event to which we can
associate a probability, proportional to the overlap between the incoming
photon and the produced vector meson wave functions. We have shown that this
simple procedure allowed us to get an excellent agreement with HERA data,
without adjustment of any additional parameters. We have noted that dipole
size fluctuations could be interpreted as fluctuations of the saturation
momentum scale of the proton. However, the phenomenological study presented in
this paper offers a simple and unified picture of the role of geometrical
fluctuations associated to natural degrees of freedom, namely the valence
quarks of the proton and the (heavy) quark components of the light cone wave
functions of the photon and the $J/\Psi$ meson.

###### Acknowledgements.

M.T. thanks the members of the Institut de Physique Théorique, Université
Paris-Saclay (CEA), for their warm hospitality during a visiting period when
the present study was initiated. He thanks also the Physics Department of
Valencia University for support and friendly hospitality. J.P.B. thanks the
ECT* in Trento for hospitality during a visit that allowed this project to be
completed.

##  Appendix A  Integration over impact parameter

We consider here the Gaussian approximation, with the proton profile given by
Eq. (28), $T_{G}({\bf b})=1/(2\pi B_{G})e^{-{\bf b}^{2}/(2B_{G})}$. The
generalization to other cases is straightforward. The Fourier transform (38)
of the dipole cross section reads (ignoring here the $z$ dependence which
factorizes)

$\displaystyle\Sigma(x_{\\!I\\!\\!P},{\bm{r}},\mbox{\boldmath$\Delta$})$
$\displaystyle=$ $\displaystyle\int d^{2}{\bf b}\,{\rm e}^{-i{\bf
b}\cdot{\mbox{\boldmath$\Delta$}}}\,{{d\sigma_{q\bar{q}}\over d^{2}{\bf
b}}}({\bf b},{\bf r},x_{x_{\\!I\\!\\!P}})$ $\displaystyle=$ $\displaystyle
2\int d^{2}{\bf b}\,{\rm e}^{-i{\bf
b}\cdot{\mbox{\boldmath$\Delta$}}}\,\,\bigg{[}1\\!-\\!\exp\left(\\!-{1\over
2}\sigma_{\rm dip}(x_{\\!I\\!\\!P},{\bm{r}})\,T_{G}({\bf
b})\\!\right)\\!\bigg{]}$ $\displaystyle=$ $\displaystyle 2\int d^{2}{\bf
b}\,{\rm e}^{-i{\bf
b}\cdot{\mbox{\boldmath$\Delta$}}}\,\,\bigg{[}1\\!-\\!\exp\left(\\!-a(x_{\\!I\\!\\!P},r)\,{1\over
2\pi B_{G}}e^{-{{\bf b}^{2}/(2B_{G})}\\!}\right)\\!\bigg{]}$

with $a(x_{\\!I\\!\\!P},r)={\pi^{2}\over 2N_{c}}{\bf
r}^{2}\alpha_{S}(\mu^{2}(r))\,{x_{\\!I\\!\\!P}}g({x_{\\!I\\!\\!P}},\mu^{2}(r))=\sigma_{\rm
dip}/2$. After expanding $\big{[}1-\exp(...)\big{]}$ in powers of $a$, one
performs the integration over the impact parameter and obtains

$\displaystyle\Sigma(x_{\\!I\\!\\!P},r,\mbox{\boldmath$\Delta$}^{2})$
$\displaystyle=$ $\displaystyle-2\sum_{n=1}^{\infty}\,{1\over
n!}\,{[-a(x_{\\!I\\!\\!P},r)]^{n}\over[2\pi B_{G}]^{n}}\,\int d^{2}{\bf
b}\,\,e^{-{{n\over 2B_{G}}{\bf b}^{2}}}\,e^{-i{\bf
b}\cdot{\mbox{\boldmath$\Delta$}}}$ (50) $\displaystyle=$
$\displaystyle-2\cdot(2\pi B_{G})\sum_{n=1}^{\infty}{1\over n}{1\over
n!}\,{[-a(x_{\\!I\\!\\!P},r)]^{n}\over[2\pi B_{G}]^{n}}\,e^{-{{B_{G}\over
2n}{\mbox{\boldmath$\Delta$}}^{2}}}$ $\displaystyle\approx$
$\displaystyle-2\cdot(2\pi B_{G})\sum_{n=1}^{n_{max}}{1\over n}{1\over
n!}\,{[-a(x_{\\!I\\!\\!P},r)]^{n}\over[2\pi B_{G}]^{n}}\,e^{-{{B_{G}\over
2n}{\mbox{\boldmath$\Delta$}}^{2}}}.$

In the last line, the sum has been truncated at a maximum value ${n_{max}}$
(for the $J/\Psi$ meson the choice $n_{max}=4$ ensures an excellent
convergence).

##  Appendix B  Overlap functions $[{\Psi^{*}_{V}\Psi_{\gamma}}]_{T,L}$

In this paper we use the same wave functions as in our previous work Traini
and Blaizot (2019). We just recall here the form of the relevant overlap for
the photo-production of $J/\Psi$ mesons (see also Kowalski  _et al._ (2006);
Forshaw  _et al._ (2004) for more details). We have

$\displaystyle[\Psi^{*}_{V}\Psi_{\gamma}]_{({\bf
r},z)}={e\over\pi\sqrt{2}}{N_{c}\over z(1-z)}\left\\{m^{2}_{f}K_{0}(\epsilon
r)\phi_{T}(r,z)\right.$
$\displaystyle\qquad\qquad\qquad-\left.\left[z^{2}+(1-z)^{2}\right]\epsilon
K_{1}(\epsilon r)\partial_{r}\phi_{T}\right\\}$ (51)

where $r=|{\bf r}|$, $e=\sqrt{4\pi\alpha_{\rm em}}\approx\sqrt{4\pi/137}$,
$\epsilon=m_{f}$ the charm quark mass, and $N_{c}=3$ is the number of colors.
Finally $K_{0}$ is the modified Bessel function of second kind, and
$\partial_{r}K_{0}(\epsilon r)=-\epsilon K_{1}(\epsilon r)$. The boosted
Gaussian wave function in configuration space is written as follows Forshaw
_et al._ (2004); Nemchik  _et al._ (1994, 1997)

$\displaystyle\phi_{T}(r,z)={\cal N}_{T}\,z(1-z)\exp\left[-{m^{2}_{f}{\cal
R}^{2}\over 8z(1-z)}+{m^{2}_{f}{\cal R}^{2}\over
2}\right]\exp\left[-{2z(1-z)r^{2}\over{\cal R}^{2}}\right]$ (52)

and ${\cal N}_{T}$ and ${\cal R}$ are fixed by normalization conditions and
the decay width Kowalski  _et al._ (2006). The values of the relevant
parameters are ${\cal N}_{T}=0.578$, ${\cal R}^{2}=2.3$ Gev-2 and $m_{f}=1.4$
Gev.

##  Appendix C Sampling procedure for $r$-Fluctuations

In order to sample the transverse overlap function displayed in Fig. 3, we
first express it in a form which is mathematically convenient for the
statistical sampling. This is achieved by a fit of the overlap with the sum of
two Gaussian functions:

$\displaystyle{2\pi r}\int{dz\over 4\pi}\,[\Psi^{*}_{V}\Psi_{\gamma}]_{(r,z)}$
$\displaystyle=$ $\displaystyle
a_{1}e^{-(r-b_{1})^{2}/c_{1}^{2}}+a_{2}e^{-(r-b_{2})^{2}/c_{2}^{2}}$ (53)
$\displaystyle=$ $\displaystyle f_{1G}(r)+f_{2G}(r)\approx F_{G}^{(2)}(r),$

with parameters given in table 1.

Table 1: Numerical values of the parameters in Eq. (53) $a_{1}$ | $b_{1}$ | $c_{1}$ | $a_{2}$ | $b_{2}$ | $c_{2}$
---|---|---|---|---|---
0.01114 | 0.07989 | 0.08078 | 0.0105 | 0.1825 | 0.1784

After normalisation, one obtains a probability distribution

$\displaystyle{\cal P}_{G}^{(2)}(r)$ $\displaystyle=$
$\displaystyle{1\over\int drF_{G}^{(2)}(r)}\,F_{G}^{(2)}(r)$ $\displaystyle=$
$\displaystyle a{1\over\sqrt{\pi c_{1}^{2}}}e^{-{(r-b_{1})^{2}\over
c_{1}^{2}}}+(1-a){1\over\sqrt{\pi c_{2}^{2}}}e^{-(r-b_{2})^{2}\over
c_{2}^{2}};$

where $a\approx 0.32$.

The sum of the two Gaussian functions in Eq. (C) can then be sampled by a
random selection of a single term of the sum, followed by the sampling of the
distribution associated to that term Kalos and Whitlock (2008).

##  Appendix D Phenomenological corrections

The derivation of the amplitude for the exclusive vector meson production (or
DVCS amplitude if $V\to\gamma$, real photon) relies on the assumption that the
amplitude ${\cal A}$ of Eq.(1) purely imaginary. The corrections due to the
presence of the real part is accounted for by the factor $(1+\beta^{2})$
multiplying the differential cross sections (2). $\beta$ is the ratio of real
and imaginary parts of the amplitude and is calculated as follows Kowalski
_et al._ (2006)

$\beta=\tan{\pi\lambda\over 2};\;\;\;\;\;{\rm
with}\;\;\;\;\lambda\equiv{\partial\ln({\rm Im}\,{\cal
A}_{T,L}^{\gamma^{*}p\to Vp})\over\partial\ln(1/x_{x_{\\!I\\!\\!P}})}.$ (56)

In addition for vector meson production (or DVCS) one should use the off-
diagonal (or generalized) gluon distributions Shuvaev  _et al._ (1999). Such
a“skewness” effect is accounted for (in the limit of small
$x_{x_{\\!I\\!\\!P}}$), by multiplying the gluon distribution
$x_{\\!I\\!\\!P}g(x_{\\!I\\!\\!P},\mu^{2})$ by a factor $R_{g}$ given by
Kowalski  _et al._ (2006)

$\displaystyle R_{g}(\lambda_{g})$ $\displaystyle=$
$\displaystyle{2^{\lambda_{g}+3}\over\sqrt{\pi}}{\Gamma(\lambda_{g}+5/2)\over\Gamma(\lambda+4)},$
$\displaystyle{\rm with}\;\;\;\;\lambda_{g}$ $\displaystyle\equiv$
$\displaystyle{\partial\ln[x_{x_{\\!I\\!\\!P}}g(x_{x_{\\!I\\!\\!P}},\mu^{2})]\over\partial\ln(1/x_{x_{\\!I\\!\\!P}})}.$
(57)

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# Identifying the atmospheric drivers of drought and heat using a smoothed
deep learning approach

Magdalena Mittermeier
Department of Geography
LMU Munich
<EMAIL_ADDRESS>
&Maximilian Weigert
Statistical Consulting StaBLab
Department of Statistics
LMU Munich
<EMAIL_ADDRESS>
&David Rügamer
Department of Statistics
LMU Munich
<EMAIL_ADDRESS>

###### Abstract

Europe was hit by several, disastrous heat and drought events in recent
summers. Besides thermodynamic influences, such hot and dry extremes are
driven by certain atmospheric situations including anticyclonic conditions.
Effects of climate change on atmospheric circulations are complex and many
open research questions remain in this context, e.g., on future trends of
anticyclonic conditions. Based on the combination of a catalog of labeled
circulation patterns and spatial atmospheric variables, we propose a smoothed
convolutional neural network classifier for six types of anticyclonic
circulations that are associated with drought and heat. Our work can help to
identify important drivers of hot and dry extremes in climate simulations,
which allows to unveil the impact of climate change on these drivers. We
address various challenges inherent to circulation pattern classification that
are also present in other climate patterns, e.g., subjective labels and
unambiguous transition periods.

##  1 Introduction

In recent summers such as those of 2003, 2010 and 2018, Europe has been
subject to particularly outstanding summer drought and heat events, which
caused large economic and societal damage including heat-related deaths [1,
2]. The frequency and intensity of hot and dry extremes has recently increased
and is projected to further increase due to climate change and rising global
mean temperatures [3, 4].

### Drivers of hot and dry extremes

There are two key processes leading to drought and heat events: thermodynamic
and dynamic factors. Thermodynamic factors involve, e.g., evaporation and the
feedback between soil moisture and air temperature. Dynamic factors on the
other hand describe the atmospheric drivers of heat and drought, which are
mainly anticyclonic conditions and blocking [3]. While anticyclonic conditions
go along with various high-pressure systems, blocking describes a particular,
persistent high-pressure situation that is associated with the displacement of
westerly winds and their accompanying weather systems [3, 5]. These
atmospheric drivers of hot and dry extremes are part of the large-scale
atmospheric circulation in the mid-latitudes, which control the weather and
climate over Europe [6, 7]. Changes in the atmospheric circulation are complex
with opposing processes and thus many open research questions remain [5, 8].

### Classification of circulation patterns

Objectively classifying the circulation patterns that are associated with hot
and dry extremes is an important step towards a better understanding of how
climate change affects the atmospheric drivers of potentially disastrous
extreme events. Previous studies, e.g., [9, 10, 11] have shown that deep
learning approaches are powerful tools for the detection of extreme weather in
climate simulations. In this study, we use a subjective catalog of circulation
type classifications over Europe by Hess & Brezowsky [12, 13]. Our goal is to
learn the categorization of six circulation types with anticyclonic conditions
over Europe, which are associated with dry and hot summer conditions in
Central Europe [14]. The classification of circulation types comes with
various challenges that need to be addressed with appropriate modeling
strategies. Challenges include noisy labels due to subjective expert choices
in ambiguous climate situations [15], an imbalanced class distribution of
labels, undefined transition days between successive circulation patterns, and
a fixed dwell time of a circulation pattern of at least three consecutive days
by its definition [13].

### Our contribution

In this work we propose a novel modeling procedure to address existing
challenges in classifying anticyclonic circulation patterns. Especially in
times of large ensembles of climate simulations [16] that consist of dozens of
model runs and thousands of model years, our study can help to efficiently
analyse large climate simulations and be another piece of the puzzle to better
understand changes in the atmospheric drivers of drought and heat.

##  2 Data

The Hess & Brezowsky catalog contains a subjective categorization of
circulation patterns created by experts manually labelling air pressure
patterns over Europe into 29 classes. In this way, daily air pressure
constellations are retrospectively assigned to one of these classes. A
circulation pattern is, by definition, required to last at least three days.
The six circulation patterns associated with heat and drought are
(abbreviations originate from German): Zonal ridge across Central Europe (BM),
Norwegian Sea-Iceland high, anticyclonic (HNA), North-easterly anticyclonic
(NEA), Fennoscandian high, anticyclonic (HFA), Norwegian Sea-Fennoscandian
high, anticyclonic (HNFA), and South-easterly anticyclonic (SEA) [17, 14]. The
mean air pressure patterns for the six circulation patterns of interest are
given in Figure 1 for the variables _sea level pressure_ and _geopotential
height at 500 hPa_ (average values at roughly 5500 meters height). For the
analysis of heat and drought, the remaining 23 circulation types are assigned
to a residual class. The frequencies of the six anticyclonic patterns are
between 8.5% (BM) and 1.4% (HNFA), whereas the residual class comprises about
80% of the days.

Figure 1: Mean air pressure patterns of the six anticyclonic circulation types
BM, HNA, NEA, HFA, HNFA, and SEA (columns) averaged over all days in the
period between 1900 and 2010. The plots are shown for the variables (rows) sea
level pressure [hPa] and geopotential height at 500 hPa [m].

Next to the catalog by Hess & Brezowsky, we supplement our data base with the
ERA-20C reanalysis data by the European Centre for Medium-Range Weather
Forecasts [18]. The long record ERA-20C data set contains global spatial
information on various climate parameters from 1900 to 2010. In accordance
with the definition by [13], we use the two atmospheric variables, sea level
pressure and geopotential height at 500 hPa as predictors for circulation
patterns. The spatial domain of our data set is defined over a region covering
Europe and the North Atlantic [13] (see Figure 1) with a spatial resolution of
5°x 5° resulting in a grid of 16 x 29 pixels. The resulting data set contains
daily information over 111 years, i.e., T = 40541 observations.

##  3 Methods

### Model definition, training, tuning and evaluation

To account for the spatial information and the specific characteristics of
circulation patterns, we use a convolutional neural network (CNN) following
[19], who propose a network architecture for climate pattern detection
problems. Since air pressure patterns of atmospheric features are
comparatively simple, our chosen architecture consists of only two
convolutional layers with larger kernels (5x5-8 and 5x5-16), a dropout layer
and two fully-connected layers as well as individual channels for both climate
parameters. The two different atmospheric variables are included as individual
channels in the CNN as in [2, 10]. While there is reason to believe that
accounting for the temporal structure of our data, e.g., through a structured
model [20] or ConvLSTM [21] improves the model, a previous study [22] showed
no improvement in the classification of circulation patterns when using a
temporal-aware architecture. As explained in the next paragraph, our approach
instead smoothes predicted labels to account for their temporal nature.

The model is trained using Adam optimization with a batch size of 128 for 35
epochs and early stopping based on a validation set of size 3650 with patience
of 6 epochs. Hyperparameter tuning for learning rate and dropout rate is
performed using Bayesian optimization [23]. We evaluate the model using
overall accuracy and macro F1-score. For class-specific evaluations, we
consider recall and precision. To obtain performance estimates that are as
unbiased as possible, a nested cross-validation with ten inner and eleven
outer folds is used. In order to not leak intra-year information, observations
within the same year are required to belong to the same fold.

### Modeling challenges

Our approach takes into account several data-specific characteristics for
circulation pattern data. First, we employ a loss-weighting scheme to account
for imbalanced classes by weighting the classes with their inverse
frequencies. Moreover, the assigned categories in the Hess & Brezowsky catalog
can be noisy, in particular for transition days between two subsequent
circulation patterns. This is due to the continuous movement of pressure
systems while circulation types are discrete by definition and in-between
states do not fit in one or the other class. We address this problem by using
label-smoothing [24] for the first and last day of each occurrence of a
specific circulation pattern. Finally, our target variable must adhere to the
aforementioned definition of a circulation pattern, implying a pattern to last
at least three days. A transition-smoothing step ensures that this three day
rule is respected. In this step, the final predicted class $\tilde{y}_{t}$ at
time point $t=2,...,T-2$, is given by

$\tilde{y}_{t}=\begin{cases}\hat{y}_{t-1}&\text{if
}\hat{y}_{t-1}=\hat{y}_{t+1}\text{ (Neighborhood Consistency)},\\\
\hat{y}_{t-1}&\text{if
}\hat{y}_{t}=\hat{y}_{t+1}\land\hat{y}_{t-1}=\hat{y}_{t+2}\text{ (2-days
Consistency)},\\\ m(\hat{\bm{\pi}}_{t-1},\hat{\bm{\pi}}_{t+1})&\text{if
}\hat{y}_{t}\neq\hat{y}_{t+1}\land\hat{y}_{t-1}\neq\hat{y}_{t+1}\text{
(Transition Membership)},\\\
m(\hat{\bm{\pi}}_{t-1},\hat{\bm{\pi}}_{t+2})&\text{else},\\\ \end{cases}$

where $\hat{\bm{\pi}}_{t}$ denotes the predicted probability vector at time
$t$, $\hat{y}_{t}=\operatorname*{arg\,max}\hat{\bm{\pi}}_{t}$ the predicted
class prior to the transition-smoothing step, and

$m(\bm{\pi}_{s},\bm{\pi}_{t})=\operatorname*{arg\,max}\\{\bm{\pi}_{u^{\ast}}\\}\text{
with
}u^{\ast}=\operatorname*{arg\,max}_{u\in\\{s,t\\}}\\{\max(\bm{\pi}_{u})\\}.$

This guarantees consistency with the required three day rule and
systematically replaces isolated single or two day-type predictions.

##  4 Results

Taking into account the aforementioned subjectivity of the circulation pattern
catalog and the noisy labels, the overall performance of the proposed model is
satisfactory and our proposed smoothing approaches consistently improve the
model across all classes. Our smoothed convolutional neural network classifier
achieves an macro F1-score of 38.4% and an overall accuracy of 59.9% averaged
over the test sets during nested cross-validation. Table 1 shows the
corresponding confusion matrix together with the precision and recall. The
best performance in terms of recall is achieved for the circulation patterns
HNA and BM, the lowest performance for NEA. In absolute numbers,
misclassifications mainly occur for residual class observations. Due to the
proposed error weighting technique, we obtain larger recall than precision
values except for the residual class.

Table 1: Confusion matrix of our proposed smoothed approach, averaged over the test sets in the nested cross-validation. Correctly classified classes are highlighted in bold. |  | LABELS |  |
---|---|---|---|---
|  | BM | HNA | HFA | NEA | SEA | HNFA | Residual | $\sum$ | Precision
OUTPUTS |  |  |  |  |  |  |  |  |  |
BM | 208.8 | 4.0 | 7.7 | 6.5 | 1.7 | 5.0 | 477.4 | 711.2 | 0.29
HNA | 11.9 | 75.8 | 3.7 | 4.5 | 7.1 | 3.7 | 204.9 | 311.5 | 0.24
HFA | 22.9 | 3.5 | 41.9 | 14.6 | 3.6 | 1.5 | 138.6 | 226.6 | 0.18
NEA | 10.4 | 2.8 | 10.4 | 61.5 | 6.3 | 11.7 | 85.9 | 188.9 | 0.33
SEA | 3.2 | 11.6 | 3.6 | 15.0 | 25.2 | 5.5 | 78.4 | 142.5 | 0.18
HNFA | 9.1 | 3.8 | 2.4 | 20.5 | 5.5 | 44.7 | 185.7 | 271.7 | 0.16
Residual | 43.3 | 7.0 | 6.6 | 5.2 | 2.7 | 5.3 | 1729.9 | 1800 | 0.96
$\sum$ | 309.6 | 108.6 | 76.2 | 127.7 | 52.1 | 77.4 | 2900.7 | 3652.4 | –
| Recall | 0.67 | 0.70 | 0.55 | 0.48 | 0.48 | 0.58 | 0.60 | – | –

Table 2 shows an ablation study of our modeling procedure. Including label-
and transition-smoothing improves the overall accuracy by 4 percentage points
and the macro F1-score by 2 percentage points compared to a model without any
smoothing steps. The class specific F1-scores also considerably increase for
all patterns. A comparison of the networks without label-smoothing and without
transition-smoothing indicates that the label-smoothing step has rather little
impact while the proposed transition-smoothing is the key to our observed
performance gains.

Table 2: Comparison of class-specific F1-scores (first 7 columns), accuracy and macro F1-Score (last two columns) for the final smoothed model (Final), a model without label-smoothing (NO LS), a model without transition-smoothing (No TS) and a model without label-smoothing (No LS and TS). Best results are highlighted in bold. |  | BM | HNA | HFA | NEA | SEA | HNFA | Residual | Accuracy | F1-score
---|---|---|---|---|---|---|---|---|---|---
MODEL |  |  |  |  |  |  |  |  |  |
Smoothed network | 0.41 | 0.36 | 0.28 | 0.39 | 0.26 | 0.26 | 0.74 | 0.60 | 0.38
No LS | 0.41 | 0.36 | 0.28 | 0.40 | 0.26 | 0.26 | 0.74 | 0.60 | 0.39
No TS | 0.39 | 0.33 | 0.25 | 0.36 | 0.23 | 0.23 | 0.70 | 0.56 | 0.36
No LS and TS | 0.39 | 0.33 | 0.25 | 0.37 | 0.23 | 0.23 | 0.70 | 0.56 | 0.36

##  5 Conclusion and Outlook

Our results indicate the high potential of deep learning-based methods in
classifying the atmospheric drivers of drought and heat. We also demonstrate
the effectiveness of our smoothed approach to deal with typical challenges in
circulation type classifications, e.g., transition-smoothing for historical
dwell time definitions. To the best of our knowledge, we are the first to use
air pressure patterns over Europe to classify circulation patterns associated
with drought and heat as given in the Hess & Brezowsky catalog. While the
proposed approach can potentially also be used for other circulation patterns
associated with different kinds of extreme climate events, our goal was to
establish a baseline model for this specific and highly relevant circulation
pattern categorization. Although our approach relies on a network architecture
developed for climate applications [19], there is room for improvement in
modeling the analyzed patterns. As an alternative to the transition-smoothing
step, we will investigate a deep hidden Markov model that accounts for the
state dwell times by assuming a latent process that emulates the data-inherent
three day transition rule.

## Acknowledgements

We thank the anonymous reviewers for their constructive comments which helped
us to improve the manuscript. The work of MM is funded through the ClimEx
project (www.climex-project.org) by the Bavarian State Ministry for the
Environment and Consumer Protection, the work of MW and DR by the German
Federal Ministry of Education and Research (BMBF) under Grant No. 01IS18036A.
The provision of the Hess & Brezowsky catalog for the years 1900-2010 by the
German Weather Service is highly appreciated. The authors of this work take
full responsibility of its content.

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