[a]Matthew Black

# Using Gradient Flow to Renormalise Matrix Elements for Meson Mixing and
Lifetimes

Robert Harlander  Fabian Lange  Antonio Rago
Andrea Shindler  Oliver Witzel

###### Abstract

Neutral meson mixing and meson lifetimes are theory-side parametrised in terms
four-quark operators which can be determined by calculating weak decay matrix
elements using lattice Quantum Chromodynamics. While calculations of meson
mixing matrix elements are standard, determinations of lifetimes typically
suffer from complications in renormalisation procedures because dimension-6
four-quark operators can mix with operators of lower mass dimension and,
moreover, quark-line disconnected diagrams contribute.

We present work detailing the idea to use fermionic gradient flow to non-
perturbatively renormalise matrix elements describing meson mixing or
lifetimes, and combining it with a perturbative calculation to match to the
$\overline{\rm MS}$ scheme using the .

##  1 Introduction

The phenomenology of $B$ physics has become a rich and diverse study area at
both dedicated $B$ factories such as Belle and BABAR and also more general
collider experiments such as those at the LHC [1]. Over many years, there has
been enormous efforts put in by the experimental community to increase the
precision of measurements of the decays and properties of $B$ mesons (see e.g.
Ref. [2]), and thus to fully leverage this success the precision of
theoretical predictions for these decays and properties should similarly
increase. In particular, the behaviour of neutral meson mixing provides a key
insight into CP violation in the Standard Model (SM) and can help constrain
elements of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix which
further tests the SM and aids in searches for new physics. Furthermore, the
lifetime of a particle is one of its fundamental properties and thus is of
great importance in testing the underlying theory for consistency. On the
theory side, the lifetimes of $B$ mesons are determined in the framework of
the heavy quark expansion (HQE) where the result is described in terms of a
series expansion in $1/m_{b}$ of perturbative Quantum Chromodynamics (QCD)
contributions and non-perturbative $\Delta B=0$ matrix elements; for a review,
see e.g. Ref. [3]. In fact, multiple calculations of the $\Delta B=2$ matrix
elements have already been carried out on the lattice [4, 5, 6, 7, 8] (and
also using QCD sum rules [9, 10, 11, 12, 13]). For recent overviews, see [14,
15]; in addition RBC/UKQCD and JLQCD presented preliminary results for on-
going work using an RI-SMOM renormalisation scheme [16, 17, 18]. When
considering the ratio of $B_{s}^{0}$ over $B^{0}$ mixing, the SU(3) breaking
parameter $\xi$ can be defined [19]. Since renormalisation factors and other
uncertainties cancel, the lattice determination of $\xi$ is typically more
precise [4, 5, 6, 7, 8, 20] and $\xi$ is frequently used in global CKM
unitarity triangle fits [21, 22, 23].

The history of the $\Delta B=0$ four-quark matrix elements is less thorough.
After early quenched studies [24, 25] and preliminary unquenched results [26]
in the early 2000s, the topic received little attention from the lattice
community until recently with some interest in lifetimes ratios and baryonic
decays [27]. In the meantime, there have been predictions using QCD sum rules
[28, 12]. These matrix elements present additional challenges for a lattice
calculation due to contributions from disconnected diagrams where the signal-
to-noise ratio worsens. Moreover, mixing with operators of lower mass
dimension occurs in standard renormalisation procedures.

In the following we outline a non-perturbative renormalisation scheme
utilising the gradient flow [29, 30, 31] and the short-flow-time expansion
[32, 33, 34, 35] in which operator mixing is absent. A perturbative matrix is
required to match each quantity to the $\overline{\rm MS}$ scheme, likely
circumventing some of the difficulty in calculating $\Delta B=0$ four-quark
matrix elements. The method is first tested for $\Delta F=2$ matrix elements
where results can be verified against the literature. Our approach is similar
to work by Suzuki et al. applying the short-flow-time expansion to neutral
Kaon mixing or the determination of the energy-momentum tensor [36, 37, 38].

##  2 Gradient Flow and Short-Flow-Time Expansion

The gradient flow [29, 30, 31, 32, 39, 40, 41, 42, 43] has become a well-known
tool in lattice simulations with common usage for e.g. scale setting. One
introduces an auxiliary dimension, the flow time $\tau$ [$\,\text{GeV}^{-2}$]
which acts as a UV regulator and provides a well-defined smearing of gauge and
fermion fields through the first-order differential equations

$\displaystyle\partial_{\tau}B_{\mu}(\tau,x)$ $\displaystyle={\cal
D}_{\nu}(\tau)G_{\nu\mu}(\tau,x),\quad B_{\mu}(0,x)=A_{\mu}(x),$ (1)
$\displaystyle\partial_{\tau}\chi(\tau,x)$ $\displaystyle={\cal
D}^{2}(\tau)\chi(\tau,x),\quad~{}~{}~{}~{}~{}~{}\,\chi(0,x)=q(x),$ (2)

where
$G_{\nu\mu}(\tau)=\partial_{\nu}B_{\mu}(\tau)-\partial_{\mu}B_{\nu}(\tau)+[B_{\nu}(\tau),B_{\mu}(\tau)]$
is the flowed gluon field strength tensor, ${\cal
D}_{\nu}(\tau)=\partial_{\nu}+[B_{\nu}(\tau),\cdot]$ is the flowed covariant
derivative, $A_{\mu},q$ are the regular gauge and fermion fields respectively
and $B_{\mu}(\tau),\chi(\tau)$ are those extended in the flow time. Operators
evolved along positive gradient flow time are removed of UV divergences and
are renormalised within a gradient flow (GF) scheme.

An effective Hamiltonian expressed normally as a sum of operators ${\cal
O}_{m}$ and their Wilson coefficients $C_{m}$ can be rewritten in terms of
these ‘flowed’ operators $\tilde{\cal O}_{n}(\tau)$ and similarly ‘flowed’
Wilson coefficients $\tilde{C}_{n}(\tau)$:

${\cal H}_{\rm eff}=\sum_{m}C_{m}{\cal
O}_{m}=\sum_{n}\tilde{C}_{n}(\tau)\tilde{\cal O}_{n}(\tau),$ (3)

where the flow-time dependence of the operators cancels with that of the
coefficients [35, 44, 45, 38]. In the short-flow-time expansion, one can
relate the ‘flowed’ operators to the regular ones as

$\tilde{\cal O}_{n}(\tau)=\sum_{m}\zeta_{nm}(\tau){\cal
O}_{m}+O(\tau)\implies\sum_{n}\zeta_{nm}^{-1}(\mu,\tau)\langle\tilde{\cal
O}^{\rm GF}_{n}\rangle(\tau)=\langle{\cal O}^{\overline{\rm
MS}}_{m}\rangle(\mu),$ (4)

where higher-dimensional operators are accompanied by higher powers of $\tau$
and are expected to be negligible for small $\tau$ [32, 33, 34, 35]. The
perturbatively-calculated matrix $\zeta_{nm}^{-1}(\mu,\tau)$ matches the GF
renormalised operators to the $\overline{\rm MS}$ scheme.

For a general heavy quark field $F$, we focus for now only on the bag
parameter of the $\Delta F=2$ four-quark operator

${\cal
O}_{1}=(\bar{F}\gamma_{\mu}(1-\gamma_{5})q)(\bar{F}\gamma_{\mu}(1-\gamma_{5})q).$
(5)

This is well-studied in the literature and is the only contributor to $\Delta
M$ in the SM. In the future, we will extend our study to consider the full
SUSY basis of $\Delta F=2$ four-quark dimension-six operators as well as
$\Delta F=0$.

The bag parameters are defined as ratios of the three-point matrix element of
a four-quark operator ${\cal O}_{i}$ to its vacuum insertion approximation.
For a pseudoscalar meson state $P$ with mass $m$ and decay constant $f$, the
bag parameter of ${\cal O}_{1}$ is defined, at leading order, as

$B_{1}=\frac{\langle P|{\cal O}_{1}|P\rangle}{\frac{8}{3}m^{2}f^{2}}.$ (6)

The perturbative calculations used in this work are described in [45, 46]. At
next-to-next-to-leading order (NNLO), the perturbative matching from GF to
$\overline{\rm MS}$ schemes for the bag parameter $B_{1}$ with number of
flavours $n_{f}$ is given by

$\displaystyle\zeta_{B_{1}}^{-1}(\mu,\tau)=1$
$\displaystyle+\frac{a_{s}}{4}\left(-\frac{11}{3}-2L_{\mu\tau}\right)$ (7)
$\displaystyle+\frac{a_{s}^{2}}{43200}\bigg{[}-2376-79650L_{\mu\tau}-24300L_{\mu\tau}^{2}+8250n_{f}+6000\,n_{f}\,L_{\mu\tau}$
$\displaystyle\qquad\qquad~{}+1800\,n_{f}\,L_{\mu\tau}^{2}-2775\pi^{2}+300\,n_{f}\,\pi^{2}-241800\log
2$ $\displaystyle\qquad\qquad~{}+202500\log 3-110700\,{\rm
Li}_{2}\left(\frac{1}{4}\right)\bigg{]},$

where $L_{\mu\tau}=\log(2\mu^{2}\tau)+\gamma_{E}$ and $a_{s}=\alpha_{s}/\pi$.
The final result for $B_{1}$ in the $\overline{\rm MS}$ scheme is given by

$B_{1}^{\overline{\rm MS}}(\mu)=\lim_{\tau\to
0}\zeta_{B_{1}}^{-1}(\mu,\tau)B_{1}^{\rm GF}(\tau).$ (8)

##  3 Lattice calculation

We will consider six RBC/UKQCD $2{+}1$-flavour domain-wall fermion (DWF) and
Iwasaki gauge field ensembles with three lattice spacings $a\sim 0.11$,
$0.08$, $0.07\,\text{fm}$ (determined by RBC/ UKQCD [47, 48, 20]) and pion
masses $\in[267,433)\,\text{MeV}$. Light and strange quarks are simulated with
the Shamir DWF action [49, 50, 51, 52] with $M_{5}=1.8$. These ensembles are
listed in Table 1.

$\begin{array}[]{ccccccccccccc}&L&T&L_{s}&a^{-1}\\!/\\!\,\text{GeV}&am_{l}^{\text{sea}}&am_{s}^{\text{sea}}&am_{s}^{\rm
val}&M_{\pi}/\\!\,\text{MeV}&\text{\\# cfgs}&\text{\\# sources}\\\
\hline\cr\text{C1}&24&64&16&1.785&0.005&0.040&0.03224&340&101&32\\\
\text{C2}&24&64&16&1.785&0.010&0.040&0.03224&433&101&32\\\\[5.16663pt]
\text{M1}&32&64&16&2.383&0.004&0.030&0.02477&302&79&32\\\
\text{M2}&32&64&16&2.383&0.006&0.030&0.02477&362&89&32\\\
\text{M3}&32&64&16&2.383&0.008&0.030&0.02477&411&68&32\\\\[5.16663pt]
\text{F1S}&48&96&12&2.785&0.002144&0.02144&0.02167&267&98&24\end{array}$ Table
1: RBC/UKQCD ensembles used in the discussed simulations [53, 54, 47, 48].
$am_{l}^{\text{sea}}$ and $am_{s}^{\text{sea}}$ are the light and strange sea
quark masses and $M_{\pi}$ is the unitary pion mass. $am_{s}^{\rm val}$ are
the valence strange quark masses, set to the physical mass.

Heavy quarks are simulated using stout-smeared gauge fields [55] and the
Möbius DWF action [56], where the mass has been tuned to the physical charm on
each ensemble through the $D_{s}$ pseudoscalar meson [57]. Using a similar
setup as Ref. [20], all propagators are generated with Z2-noise wall sources
where the number of sources and smearing parameters are listed in Table 1;
Gaussian smearing is also applied for the strange quarks.

In the following, we use exploratory results obtained on the C1, C2, and M1
ensembles. While testing the validity of our method, we remove the additional
complications of extrapolations in the valence sector, studying only strange
and charm quarks at their physical values. As such, we currently consider the
short-distance contributions to ‘neutral $D_{s}$’ meson mixing.

On the lattice, this is obtained in the large $t$ and $\Delta T$ limit by the
ratio of correlation functions,

$R_{1}(t,\Delta T,\tau)=\frac{C_{{\cal O}_{1}}^{\rm 3pt}(t,\Delta
T,\tau)}{\frac{8}{3}C_{AP}^{\rm 2pt}(t,\tau)C_{PA}^{\rm 2pt}(\Delta
T-t,\tau)}\to B_{1}^{\rm GF}(\tau),$ (9)

where $t$ is the Euclidean time and $\Delta T$ is the separation of the two
sources used in the three-point function as shown in Figure 1, and
$C_{AP}^{\rm 2pt}(t,\tau),C_{PA}^{\rm 2pt}(\Delta T-t,\tau)$ are the two-point
functions with the pseudoscalar current at the sources and the flowed axial
current at the sink. In this pilot study we only consider $\Delta T=28$ for
all data analysed so far.

Figure 1:  Quark line diagram of the lattice setup calculating the three-point
correlation function with $\Delta F=2$ four-quark operator insertion at time
$t$ between two sources placed at $t_{0}$ and $t_{0}+\Delta T$.

Dependence on the flow time $\tau$ is written here explicitly as the above
ratio is evaluated for propagators taken at discrete steps in the flow. The
Runge-Kutta evolution of the gradient flow is performed with $\epsilon=0.01$.
For small flow, measurements of two- and three-point functions are taken at
steps of $\varepsilon=0.1$ in lattice units, with this ‘coarsening’ to
$\varepsilon=0.4$ for $\tau/a^{2}>5$.

##  4 First Results

In Figure 2, we present our first results for the $\Delta F=2$ bag parameter
$B_{1}^{\rm GF}(\tau)$ and its GF-to-$\overline{\rm MS}$ matching coefficient
at both NLO and NNLO, expressed with the renormalisation scale $\mu=3\,$GeV
both as functions of the gradient flow time $\tau$ in physical units.

Figure 2:  Evolution of the $\Delta F=2$ bag parameter $B_{1}^{\rm GF}(\tau)$
along the gradient flow for ensembles C1, C2, M1 in physical units with
statistical errors only (left); perturbative GF-to-$\overline{\rm MS}$
matching coefficient $\zeta^{-1}_{B_{1}}(\mu,\tau)$ as a function of flow time
$\tau$ with $\mu=3\,$GeV at both NLO and NNLO from Refs. [45, 46] (right).

The plot on the left shows the dependence of the lattice data on the GF time
$\tau$ converted to physical units. The clear overlap of the data from
different ensembles indicates a mild continuum limit for large enough flow
times ($\tau\gtrsim 0.2\,{\rm GeV}^{-2}$) where the results become GF
renormalised; for smaller flow times, the continuum limit may however carry a
substantial systematic uncertainty. The plot on the right shows the
perturbative matching with a clear difference between the next-to-leading
order (NLO) and NNLO results.

When combining with the data from the lattice simulations, we expect to obtain
the $\overline{\rm MS}$-renormalised result by taking the $\tau\to 0$ limit
assuming a linear dependence on the GF time. The outcome is shown in Figure 3.
The purple circles utilize the NLO matching to $\overline{\rm MS}$-scheme,
whereas the orange squares use NNLO matching coefficients. In both cases an
extended linear region is present. Using the NNLO coefficients this linear
region extends to lower flow times compared to the NLO results. The next step
is to seek a window in flow time where the gradient flow has had sufficient
effect on the lattice results such that they are renormalised but the flow
time is still small enough for higher-dimensional operators to be suppressed.

If we use as a first guess a flow time window $0.25\,{\rm
GeV}^{-2}\leq\tau\leq 0.67\,{\rm GeV}^{-2}$ for NNLO and $0.39\,{\rm
GeV}^{-2}\leq\tau\leq 0.67\,{\rm GeV}^{-2}$ for NLO, we can perform the
$\tau\to 0$ extrapolation shown by the gray bands to obtain the renormalised
bag parameter in the $\overline{\rm MS}$-scheme at zero flow time. While the
difference in the prediction may indicate systematics due to the order of the
perturbative matching, we also point out that other systematic effects, e.g.
from the continuum extrapolation, need to be accounted for. In addition we
only consider a naive error estimate for the $\tau\to 0$ limit which warrants
improvement.

Similar discussions regarding the $\tau\to 0$ extrapolation for the energy-
momentum tensor in the short-flow-time expansion framework are given in e.g.
Refs. [36, 37]. Further study is still required to fully understand the
validity range of both extrapolations.

Figure 3:  Results for the combination $\zeta^{-1}_{B_{1}}(\mu,\tau)B_{1}^{\rm
GF}(\tau)$ as a function of flow time, taking NLO (purple) and NNLO (orange)
perturbative matching with $\mu=3\,$GeV. Our error bars reflect only
statistical uncertainties but we know that for small flow times ($\tau\lesssim
0.2\,{\rm GeV}^{-2}$) the continuum extrapolation contributes a large
systematic uncertainty. Using the purple/orange shaded region in flow time, we
perform an uncorrelated fit for the $\tau\to 0$ extrapolations shown as gray
bands.

Phenomenologically, ‘neutral $D_{s}$’ mixing as is calculated here does not
exist, however the results should be similar in magnitude to that of short-
distance $D^{0}$ mixing since any spectator effects are expected to be small.
In the literature, the short-distance matrix elements for $D^{0}$ mixing have
been calculated on the lattice by FNAL/MILC at $N_{f}=2+1$ and ETMC at
$N_{f}=2+1+1$, with $\mu=3\,$GeV. ETMC finds a value of $B_{1}^{\overline{\rm
MS}}=0.757(27)$ [58] (they also have a calculation at $N_{f}=2$ [59]).
FNAL/MILC quotes a value for $\langle{\cal O}_{1}\rangle^{\overline{\rm MS}}$;
using PDG 2023 [57] and Eq. (6), this leads to $B_{1}^{\overline{\rm
MS}}=0.795(56)$ [60]. In Ref. [12], there also exists a QCD sum rules
calculation which, using PDG 2023 [57], results in $B_{1}^{\overline{\rm
MS}}=0.636^{+0.091}_{-0.079}$. One can see in Figure 3 that our preliminary
results extracted here lie between the two literature values from lattice QCD
and slightly above that from QCD sum rules. While further scrutiny is still
required, this is a promising sign for our method as a novel renormalisation
and matching-to-$\overline{\rm MS}$ procedure. It motivates further study of
$\Delta F=2$ matrix elements in the short-flow-time expansion as a test case
towards a calculation of the long-sought-after $\Delta B=0$ four-quark matrix
elements.

##  5 Summary

The $\Delta B=0$ four-quark dimension-six matrix elements are important
quantities in accurately and precisely predicting the lifetime of a $B$ meson
from the heavy quark expansion. Lattice QCD calculations of these matrix
elements are strongly sought-after, but no full calculation has been performed
to date, with part of the difficulty coming from mixing with lower-dimensional
operators under standard renormalisation procedures. Here we have outlined the
idea of using the gradient flow and short-flow-time expansion as an
alternative renormalisation scheme and matching-to-$\overline{\rm MS}$ method
to bypass the issue of operator mixing. First simulations were carried out
with the focus on $\Delta F=2$ operators where results can be validated
against lattice calculations in the literature. Removing additional
extrapolations, the initial analysis has been performed at the physical
$D_{s}$ scale. Preliminary results show promise and consistency with
literature values of short-distance contributions to $D^{0}$ mixing. However
further scrutiny on estimating systematic uncertainties is warranted and
getting deeper insight in how to choose the flow time window is desired.

In future work, we aim to extend simulations to all lattice ensembles listed
in Table 1, and also to multiple heavy quark masses and replacing the strange
quarks with light quarks. This will allow extrapolation to physical $B$ and
$B_{s}$ systems where further validation against $\Delta B=2$ calculations may
be done and physical results for the ultimate goal of the $\Delta B=0$ matrix
elements can be reached.

## Acknowledgments

Measurements were performed using Grid [61, 62] and Hadrons [63]. Computations
used resources provided by the OMNI cluster at the University of Siegen and
the HAWK cluster at the High-Performance Computing Center Stuttgart. This work
was partially supported by DeiC National HPC (g.a. DEIC-SDU-L5-13). We used
gauge field configurations generated on the DiRAC Blue Gene Q system at the
University of Edinburgh, part of the DiRAC Facility, funded by BIS National
E-infrastructure grant ST/K000411/1 and STFC grants ST/H008845/1, ST/K005804/1
and ST/K005790/1. M.B., R.H., F.L., O.W. received support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through grant
396021762 - TRR 257 “Particle Physics Phenomenology after the Higgs
Discovery”. The work of F.L. was supported by the Swiss National Science
Foundation (SNSF) under contract TMSGI2_211209. We thank ECT* for support at
the Workshop “The Gradient Flow in QCD and other strongly coupled field
theories” during which this work has developed. Special thanks is given to
Felix Erben, Ryan Hill, and J. Tobias Tsang for assistance in setting up the
simulation code.

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# Taming Aerial Communication with
Flight-assisted Smart Surfaces in 6G Era

Francesco Devoti, Placido Mursia,
Vincenzo Sciancalepore, Xavier Costa-Pérez F. Devoti, P. Mursia, and V.
Sciancalepore are with NEC Laboratories Europe GmbH, 69115 Heidelberg, Germany
(e-mails: {francesco.devoti, placido.mursia,
vincenzo.sciancalepore}@neclab.eu).X. Costa-Pérez is with NEC Laboratories
Europe GmbH, Heidelberg, Germany, and i2CAT Foundation and ICREA, Barcelona,
Spain. (e-mail: xavier.costa@ieee.org).This work was supported by the European
Union H-2020 Project RISE-6G under grant 101017011.

###### Abstract

Aerial communication is gradually taking an assertive role within common
societal behaviors by means of unmanned aerial vehicles (UAVs), high-altitude
platforms (HAPs), and fixed-wing aircrafts (FWAs). Such devices can assist
general operations in a diverse set of heterogeneous applications, such as
video-surveillance, remote delivery and connectivity provisioning in crowded
events and emergency scenarios. Given their increasingly higher technology
penetration rate, telco operators started looking at the sky as a new
potential direction to enable a three-dimensional (3D) communication paradigm.

However, designing _flying mobile stations_ involves addressing a daunting
number of challenges, such as an excessive on-board control overhead, variable
battery drain and advanced antenna design. To this end, the newly-born _Smart
Surfaces_ technology may come to help: _reconfigurable intelligent surfaces
(RIS)_ may be flexibly installed on-board to control the terrestrial
propagation environment from an elevated viewpoint by involving low-complex
and battery-limited solutions. In this paper, we shed light on novel RIS-based
use-cases, corresponding requirements, and potential solutions that might be
adopted in future aerial communication infrastructures.

##  I Introduction

Aerial communications, whereby flying devices such as unmanned aerial vehicles
(UAVs), fixed-wing aircrafts (FWAs) or high-altitude platforms (HAPs) are used
to facilitate the communication between two given nodes in the network, are a
well-established technology capable of providing powerful yet highly flexible
platforms to enhance and complement current fifth generation (5G) and beyond
network infrastructures (6G) [1]. Such kind of deployments, denoted as _air-
to-ground (A2G)_ networks, are envisioned as an integral part of future
internet of things (IoT) networks thanks to their agility to act as portable
on-demand base stations (BSs), flexibility to move in space, and an overall
increased probability of high end-to-end channel quality. As a matter of fact,
unlike fixed BSs, flying devices can be exploited on-demand and reused for
different applications, e.g., assisting rescue operations after natural
disasters via user localization and (re)establishing a damaged or unavailable
network infrastructure [2]. In addition, they are able to hover avoiding
obstacles that may cause blockage. Thus, significantly increasing the
probability of establishing a line-of-sight (LoS) with terrestrial users,
giving rise to the new three-dimensional ($3$D) communication paradigm [3] in
the 6G era.

However, despite its potentials and envisioned business opportunities, aerial
communications are hindered by the limited power budget and maximum carrying
load of flying devices. E.g., in the case of UAVs it poses significant
constraints on its feasibility: carrying bulky, heavy, and power-hungry
equipment such as active antennas on board results in high design complexity
and capital expenditure; further limiting the application range of aerial
communications. Moreover, conventional A2G networks require signal processing
capabilities on-board, e.g., for channel estimation or precoding, and frequent
backhaul communications, which further compound the design complexity of such
systems.

Figure 1: RIS-based $3$D communication environment with UAV s, HAP s and FWA
s.

To overcome the above-mentioned issues, there has been growing interest in the
emerging reconfigurable intelligent surface (RIS) technology, which is widely
recognized as the means to greatly improve the quality of wireless
communication links towards 6G networks (A2G communications) as depicted in
Fig. 1 [4]. RISs are two-dimensional surfaces divided into unit cells spaced
at a sub-wavelength distance, which can reflect the incoming signal by adding
a given tunable phase shift. As a result, the reflected signal components can
be combined constructively at the intended receiver position and destructively
elsewhere in a nearly-passive way [5]. Moreover, RISs can be fabricated to be
extremely lightweight (hundreds of grams) and consume as little as tens of
milliwatts, as opposed to conventional BSs that weight few Kilograms and
consume tens of Watts. Thereby, making them suitable for being rapidly
deployed in B5G/6G networks at affordable costs.

While the opportunities of RIS-aided A2G networks are well-understood, several
design aspects still need to be addressed in order to unlock its full
potential. Indeed, the joint optimization of the RIS and BS configuration, as
well as the aerial device trajectory, require an adaptive level of control and
signaling able to cope with the degree of mobility of the specific application
scenario, including: unpredictable movements typical of aerial objects,
corresponding maneuvering and a limited power budget available on board. All
such aspects pose strict limitations on the A2G control architecture, making
its design extremely challenging.

In this paper, we identify the main optimization aspects to be considered when
designing a RIS-aided A2G network by describing its main characteristics and
corresponding state of the art (SoA) solutions. We further shed light on
technical challenges and open research directions and characterize two
relevant operating scenarios. We finally deliver the message that _while joint
exploitation of RIS energy-efficient beamforming capabilities and flexible and
agile aerial communications requires a careful design of the control
architecture, it might significantly boost the overall communication
performance_.

##  II RIS-Aided A2G Networks

RISs can be integrated in the context of aerial communications in two
different ways, namely i) terrestrial RIS, i.e., by mounting them on the
facade of buildings in order to assist in the communication to/from the flying
object and ii) aerial RIS, i.e., by employing them as substitutes to bulky
active components such as conventional BSs on board the flying device [6].
However, while deploying terrestrial RISs can help alleviate the total power
consumption, it does not fundamentally solve the problem of limited operation
range of e.g. UAVs. On the other side, by allowing the UAV to carry an
inherently light-weight and passive device on board such as an RIS, nearly all
the available power can be devoted to enlarge the range of operation while
simultaneously achieving highly selective beamforming. The RIS itself might
even be (partially) powered by harvesting power from the incoming signals,
which further demonstrates the feasibility of aerial RISs [7]. Moreover,
terrestrial RISs can only achieve $180^{\circ}$ reflection angles, compared to
the full-angle $360^{\circ}$ of aerial RISs, which allows to effectively cover
the intended two-dimensional target area. Therefore, we focus on aerial RISs
as depicted in Fig. 1 and describe the main characteristics, involved
opportunities, and corresponding challenges.

The link budget of an A2G network is directly proportional to the signal
wavelength and the number of RIS elements, and inversely proportional to the
distance to the user [8]. On the one hand, the increasing demand for higher
communication rates and the need to support a massive number of users is
pushing future wireless networks towards higher frequency bands, such as
millimeter-wave (mmWave) or terahertz (THz). Hence, the signal wavelength is
typically small and will likely tend to diminish, which further compounds the
problem of correctly designing A2G networks. On the other hand, while the
received power at the ground user can be linearly increased by making RIS
elements dense, UAVs can carry only a limited payload thus resulting in a
maximum feasible RIS dimension. Note that to overcome this problem, RIS-
equipped UAV swarms can be deployed, i.e., a group of aerial RISs that
cooperate together [9]. However, such cooperation comes at a significantly
increased cost in terms of control overhead and complexity. It is thus of
paramount importance to optimize the location of the aerial platform in order
to obtain the best possible propagation conditions to the ground users in
terms of increased LoS probability and limited pathloss.

Beamforming at the RIS can increase the received power at the intended
location up to a maximum theoretical factor equal to the square of the number
of elements. In order to take full advantage of such tremendous capabilities,
advanced $3$D passive beamforming algorithms and techniques at the RIS are
needed, especially in the multi-user scenario, which requires addressing
complex non-convex problems.

Moreover, current 5G BSs are usually equipped with a multi-antenna array,
which endows them of active beamforming capabilities. The optimization of both
active and passive beamforming vectors is intrinsically coupled with the
optimization of the position of the flying device and its evolution over time
(i.e., its trajectory). Such joint optimization is in general intractable and
requires to resort to suboptimal approaches or heuristics. In particular, an
efficient approach is decoupling the two problems in an alternating fashion.
The trajectory optimization is efficiently solved via machine learning (ML)
tools such as deep networks, while the passive beamforming optimization
problem can be tackled via successive convex approximation (SCA) or
semidefinite relaxation (SDR) [10]. In the multi-user scenario, one can
alternatively schedule users with a time division multiple access (TDMA)
scheme [11] or, in order to improve fairness and robustness, consider max-min
approaches thereby assigning at each user a given figure of merit such as,
e.g., the signal-to-noise ratio (SNR) or the bit error rate (BER), and aiming
at maximizing the worst figure among the target users [12].

The core of the joint trajectory and beamforming optimization in A2G networks
is linked to the acquisition and tracking of channel state information (CSI).
Indeed, an accurate system optimization requires the estimation of both the
channel from the BS to the flying device and the channel from the latter to
the target user. However, RISs are passive structures with no estimation nor
processing capabilities, which poses several constraints on the applicability
of classical pilot-based channel estimation techniques [13]. In addition, in
real-life scenarios the users have a certain degree of mobility, which
presents the highly challenging problem of tracking the evolution of the
channel vectors over time. In this regard, it is essential to have accurate
channel modeling that depends on few key system parameters. Thanks to the high
LoS probability associated with the altitude of the aerial platform thereby
enabling the $3$D communication paradigm [3], the channel vectors are
completely characterized by the distance travelled by the signal and the angle
of arrival (AoA)/angle of departure (AoD) at the various entities in the
network [14]. It is thus possible to realize efficient CSI acquisition in A2G
networks by tracking the location of both the users and the flying device,
which is feasible with a sufficient degree of control signalling.

##  III A2G Control Architecture

As depicted on the left-hand side of Fig. 2, SoA control architectures for A2G
networks in the case of UAVs are typically divided into two separate layers:
ground control station and aerial platform control. The former is physically
located in the terrestrial network and it consists of a processing unit that,
given a set of policies and quality of service (QoS) requirements, jointly
optimizes the UAV trajectory and both the active and passive beamforming at
the BS and at the RIS, respectively. Moreover as described above, it deals
with acquiring CSI and extracting the associated relevant channel parameters.
Whereas, the aerial platform control is given by the on-board UAV controller
and the RIS controller. While the former is dedicated to the maneuvering of
the UAV, the latter triggers RIS settings (i.e., predefined phase shifts).

Figure 2: Standard communication and control framework for RIS-aided UAV
scenario, with additional control enhancement modules and interfaces enabling
advanced A2G control.

A widely adopted assumption in SoA control architectures is to consider the
UAV in a predefined location in space with negligible orientation and position
variations during the communication phase, while its position is updated only
within the displacement phase. However, such assumption does not hold in
practical scenarios, wherein the UAV maneuvering and several atmospheric
phenomena can change the position and the orientation of the UAV even during
the communication operations, leading SoA solutions to be potentially
inefficient—or even unfeasible—to operate in realistic conditions. Indeed, as
the UAV is hovering at a certain altitude, its motion is influenced by a
deterministic component, which is due to the intentional maneuvering of the
UAV, i.e., following a predefined trajectory, and a random component, due to
unpredictable factors such as atmospheric conditions including wind, rain, and
humidity, imprecise maneuvering, non-ideal UAV instrumentation, etc. Such
movements result in translations and rotations of the surface of the on-board
RIS, which in turn lead to misalignment of the transmit and reflected beams.
This effect is further exacerbated by the highly directive nature of mmWave
beamforming at the RIS and can ultimately result in loss of connectivity at
the user-side [15].

To get the most from A2G networks in a practical scenario, the mitigation of
UAV mobility effects on the QoS is a key point. It is thus essential to design
enhanced control architectures enabling a transmission optimization tightly
coupled with the UAV mobility pattern. Indeed, UAVs are equipped with
different sensors, such as gyroscope, compass, global positioning system
(GPS), etc., that provide the UAV controller with motion feedback enabling
hoovering control and stabilization. However, due to the separation of the UAV
and RIS controllers, information on UAV movements is only partially considered
during the system optimization phase, i.e., only the nominal position of the
UAV is considered to perform CSI acquisition and joint trajectory and
beamforming optimization, while information such as real-time maneuvering
instructions and UAV sensors’ output is typically neglected. Fig. 2 shows the
framework that we envision to enable interaction between the UAV and the
control architecture. In particular, on the right-hand side we consider two
distinct artificial intelligence (AI)-based modules dubbed as on-board
intelligence and in-network intelligence, which are located on the UAV, and at
the network-side, respectively. The on-board module interacts both with the
UAV and the RIS controllers. It plays the fundamental role of collecting
mobility information from the UAV and integrating it into the communication
optimization process. Interestingly, the in-network intelligence module
communicates both with the on-board module and with the standard communication
optimization module. Our proposed novel framework is thus capable of blending
together the various conventionally separated system entities and effectively
utilizing all the available precious information to suitably optimize both RIS
and BS parameters such as, e.g., beamforming configurations and transmit power
at the BS.

##  IV Key-design aspects

To describe our envisioned enhanced control architecture, we first identify
the key aspects that significantly affect its design. As depicted in Fig. 3,
an A2G network can explore three main directions: i) the total available power
budget, ii) the chosen reconfiguration rate, i.e., the rate at which the key
system parameters—such as the UAV position, the beamforming vectors at the BS
and the RIS, the CSI, etc—need to be updated, and iii) the maneuvering
control, i.e., identifying which entity in the system is controlling the UAV
and how this interacts with the rest of the network. In the following, we
describe the impact of each one of the aforementioned orthogonal directions,
which are tightly coupled with the chosen hardware to be employed at both the
UAV and the RIS, and the desired application scenario, with its associated
physical propagation conditions and requirements in terms of QoS.

Figure 3: Design choices for the A2G enhanced control architecture.

Power availability. The specific UAV hardware directly determines the
available power budget on board and the percentage used for flying or
hovering. In addition, the UAV may be proprietary of the network operator or
might be outsourced from a third-party to lower capital expenditures. In the
first case, all the available power is dedicated to optimizing the underlying
A2G network, whereas in the second case the UAV movements are independently
controlled by an external entity. Indeed, nowadays UAVs are used for a
plethora of different applications such as package delivery or surveillance
and it is thus possible to equip them with RISs that can be opportunistically
used by the network. On the other hand, the size and weight of the RIS affect
the total power consumption, while its number of antenna elements defines the
degree of computational power and overhead at the RIS controller, and at the
network-side, e.g., to determine the passive beamforming configuration. In
addition, the chosen application scenario plays a fundamental role in the
total power availability. In particular, the number of users and their
mobility behaviors, and the propagation conditions such as the presence of
obstacles or unwanted meteorological phenomena are key aspects to be carefully
considered. Moreover, the UAV needs to be re-charged in appropriate charging
stations, whose location constraints the total power availability in the
environment. Such aspects have a direct impact on the joint beamforming and
trajectory optimization of the RIS and the UAV, respectively, and thus imply a
given computational power. The latter is also directly influenced by the
required QoS: in disaster situations the focus is on sending emergency time-
critical signals that are characterized by low latency and low rate, thus
giving rise to low computational power, whereas less critical applications
such as data streaming may accept a larger latency but require a higher rate,
which is typically associated with an increased computational power. In this
regard, we identify three categorizations namely unconstrained, feasible, and
limited power availability.

Reconfiguration rate. The choice of reconfiguration rates enables different
QoS guarantees and overall achievable communication rate. We determine three
degrees of increasing reconfiguration rate dubbed as intermittent, regular,
and frequent. For low reconfiguration rates, the system can devote most of the
available time for sending/receiving data to/from the users. However, the
available information from the UAV sensors, user feedback, and the CSI
acquisition is only sporadically updated. As a result, the performance might
be acceptable only for low mobility scenarios or when the atmospheric
conditions are ideal. Whereas for frequent reconfiguration rate the available
information is continuously updated and as a result, the communication quality
can be generally maximized even under high user mobility and strong
meteorological perturbations. However, this implies an increased overhead that
negatively affects the overall data rate. An intermediate solution is given by
regular reconfiguration rates, which is the case of robust optimization
algorithms, i.e., schemes that employ statistical channel and perturbation
information rather than costly instantaneous CSI.

Maneuvering. The UAV maneuvering is either handled by the network itself,
i.e., it is coupled with the optimization of the system, or by an operator
like in disaster situations, in which the information about the UAV position
is feedback based from the control device. In such scenarios, the network can
access useful information from the UAV sensors and can accurately track the
CSI as the UAV moves. In the second case, the UAV maneuvering might be under
the control of an external entity and is thus disjoint from the rest of the
system optimization. Hence, the CSI and the UAV position must be regularly
estimated in order to keep an acceptable level of performance.

##  V RIS-aided A2G Case Studies for 6G

Figure 4: Practical scenario with static UAV subject to undesired displacement
and oscillation due to atmospheric phenomena. The output of the UAV sensors
are processed and used to obtain a CSI representation accounting for the UAV
mobility statistics. The RIS configuration is optimized accordingly to
preserve the communication quality.

Hereafter, we describe a practical implementation of our proposed enhanced
control architecture considering two relevant case studies when RIS technology
is in place.

Case study: Static UAV. The relevant scenario of interest is depicted in Fig.
4. Assume that due to low mobility of ground users a UAV is in a fixed
location in space, which is only seldomly updated. However, due to adverse
atmospheric conditions, the UAV is subject to unwanted perturbations, which
result in undesired roll, yaw, and pitch of the surface of the on-board RIS.
UAV movement counteractions are automatically taken but still orientation
oscillations or location perturbation may result in an instantaneously RIS
misconfiguration, leading to misalignment of the reflected beams and degraded
overall achievable rate. Remarkably, it has been recently shown that it is
possible to guarantee an acceptable level of performance in the target area of
influence by suitably adjusting the beamforming configuration as a function of
the second-order statistics of such perturbations [15]. Such adaptation of the
system configuration is enabled by our proposed enhanced control architecture.
Indeed, the measurements of the instantaneous roll, yaw and pitch are
collected by the UAV controller and sent to the UAV data processing module
that extracts or predicts (e.g., using AI) the relevant statistics. This
information is then used upon request to update the current CSI and optimize
the communication to/from the UAV. In particular, the beamforming
configuration can be optimized on the basis of the current perturbation
statistics via: i) conventional mathematical tools such as SDR, ii) by
training a ML model that learns how to adapt the beamforming configuration to
the varying atmospheric conditions, or iii) by designing an online AI learning
algorithm. The choice of optimization method determines the computational
power expenditure, which has a direct impact on the total power availability.
As a result, our proposed enhanced control architecture can be designed to
have a feasible or unconstrained power availability.

As the UAV is in a fixed position, the RIS and BS precoders can be updated
only when needed, i.e., when the perturbation statistics evolve due to a
change of atmospheric conditions. Our proposed enhanced control architecture
provides two ways to deal with such a scenario, which are characterized by an
intermittent and regular reconfiguration rate, respectively. A first approach
is to exploit feedback monitoring: users in the service area can periodically
send updates to the network with the current perceived QoS. The in-network
intelligence devoted to the CSI acquisition issues a statistics update request
to the UAV data processing module if relevant changes in the QoS are detected.

A second approach is based on proactively sending updates on the UAV
perturbation statistics when a relevant change is detected from the UAV data
processing module to the CSI acquisition module. The latter then decides
whether the relevant system parameters, such as the beamforming
configurations, should be updated. This approach leads to higher power
consumption at the UAV due to continuous monitoring and increased
communication overhead. On the other hand, it allows to quickly react to the
varying environmental conditions and minimize network downtime.

Lastly, note that both aforementioned approaches can be realized under coupled
or feedback-based maneuvering. Indeed, in such cases, the UAV is controlled by
the network and can thus retrieve useful information from the UAV controller.
Whereas, a disjoint implementation would make it unfeasible to extract
statistical information on the unwanted perturbations of the UAV position.

Case study: Nomadic UAV. We consider the practical case where a UAV is subject
to desired movements in space, i.e., following a predetermined trajectory, as
depicted in Fig. 5. For simplicity, we neglect user mobility in order to focus
on providing coverage enhancement within a given target service area. The
drone movements lead to a continuous position and orientation change of the
on-board RIS. Such effect, if not properly addressed, could lead to severe
beam misalignment, and potentially to complete service disruption. Therefore,
to maintain a stable connection, a continuous adaptation of both the RIS
configuration and the BS precoder is required.

Figure 5: Practical scenario with UAV subject to intentional maneuvering
operations with corresponding displacement and orientation changes. The
maneuvering operation and the output of the UAV sensors is processed and used
to obtain a prediction of the UAV position and orientation in time. The RIS
configuration is optimized according to the UAV motion to enable transmission
adaptation.

Referring to the degrees of freedom highlighted in Section IV, we consider the
UAV maneuvering to be either _coupled_ , i.e., the UAV trajectory is imposed
by decisions taken at the network-side and therefore jointly optimized with
the beamforming strategy, or _feedback-based_ , i.e., the UAV is controlled by
an external operator (e.g., a member of first responder teams) and therefore,
its movements are not perfectly known to the network.

In the case of _coupled_ maneuvering, the communication optimization module
can compute in advance the RIS and the BS beamforming configuration according
to the UAV trajectory evolution. Meanwhile, thanks to the UAV sensors data,
the on-board data processing module can track the effective trajectory
evolution, compare it against the desired one, and feed back information in
case of divergence (e.g., non-idealities of the UAV controller, wind, etc.).
Thus enabling suitable adjustment of both the beamforming and the maneuvering,
and improving the overall reliability and robustness of the system. Depending
on the power availability and variety of sensors equipping the UAV, the on-
board data processing module can be further exploited. For example, visual
information from cameras could be used to perform object detection and reveal
potential obstruction (e.g., buildings, trees, or debris) and adapt the UAV
trajectory accordingly.

Whereas in the case of _feedback-based_ maneuvering, the UAV movements are
known at the network-side only a-posteriori. Therefore, to prevent the
communication optimization from lagging behind the UAV movements, a trajectory
prediction strategy could be applied to the on-board data processing module.
Such prediction could take advantage of both the sensor data and the
maneuvering feedback, while the trajectory forecast could be sent to the
network-side to compute the optimal beamforming configuration in advance, so
as to enable a more accurate transmission adaptation to the UAV movement.

The need for continuous communication adaptation, specific of the nomadic UAV
scenario, results in a control architecture design that has to sustain a
relatively high reconfiguration rate, i.e., _regular_ or _frequent_. This
implies a potentially high control overhead to frequently transmit the RIS
configuration to the RIS controller. Nonetheless, smart overhead reduction
strategies can be implemented thanks to the proposed enhanced control
architecture. Indeed, the reconfiguration can be avoided as long as the QoS is
within the desired level, despite the movement of the UAV. Therefore, a rate
adaptation strategy could be considered to trigger the reconfiguration only
when needed, e.g. based on the users’ feedback monitoring, thus minimizing the
communication overhead. Moreover, the RIS configuration typically exhibits
regular and periodic patterns over the surface elements. This peculiarity can
be exploited by advanced encoding and decoding techniques, e.g., autoencoders,
that could be easily implemented in-network (encoding) and on-board (decoding)
to further reduce the overhead. Such an additional feature would stress the
computational load of the on-board intelligence module and in turn reflect on
the power availability requirements. As a result, this scenario is
characterized by limited or feasible power availability.

Figure 6: Comparison of the achievable rate and the control overhead in a
nomadic UAV scenario obtained with standard periodic RIS reconfiguration and
smart reconfiguration adaptation enabled by the enhanced control architecture
versus the UAV speed.

##  VI Discussion

The performance of our proposed enhanced control architecture in the
application scenario described in Section V are hereafter summarized, for the
case of feedback-based maneuvering, i.e., in the presence of an external UAV
operator. We consider an A2G network including one single-antenna BS, one UAV
equipped with a $100$ element squared RIS and a target single-antenna user
located at a distance of $70$ meters from the BS. We assume the operator moves
the UAV following a trajectory encircling the user with a radius of $25$ m,
altitude of $20$ m, and with a given variable speed. The working frequency is
set to $30$ GHz, while the transmit power at the BS is fixed to $24$ dBm and
the noise spectral density is assumed to be $-80$ dBm.

In Fig. 6, we compare two different schemes, namely our proposed enhanced
control architecture dubbed as Adaptive and the standard SoA control framework
namely Fixed. We vary the UAV speed from $5$ to $50$ km/h and consider two
different choices of reconfiguration rate, denoted as Frequent in solid lines
and Regular in dashed lines. Thanks to our proposed control architecture, the
in-network intelligent modules receive feedback from the UAV maneuvering, the
UAV sensors and the user-perceived QoS and use it to adapt the system
reconfiguration rate to the current UAV mobility rate and effective
trajectory. In particular, as shown on the right-hand side, our proposed
scheme adapts the CSI and RIS beamforming strategy more frequently for an
increasing UAV speed. On the one hand this generates higher overhead, which is
shown as a percentage of the total available transmission time. On the other
hand, the data rate is kept high and quasi-constant for the case of frequent
reconfiguration rate thanks to the up-to-date CSI, which in turn leads to high
SNR.

In contrast to this, the standard architecture does not have access to the
aforementioned UAV and QoS status information and thus, uses a pre-determined
refresh rate to perform CSI acquisition and RIS beamforming optimization, as
is typically the case for directional communications such as mmWave. In this
case, the communication overhead is constant versus the UAV speed, while the
data rate monotonically decreases. However, at low UAV speeds, the standard
scheme obtains higher data rates at the cost of an increased overhead as
compared to the proposed enhanced scheme.

##  VII Conclusions

Aerial communications are opening a new research direction to enable a $3$D
mobile networking paradigm expected to be effective in the 6G landscape.
However, a number of daunting challenges need to be addressed before the dream
of cost-effective _flying mobile stations_ solutions can be reached, in
particular with respect to the design of the control architecture.

In this paper, we analyzed the novel concept of RIS-aided aerial
communications and shed the light on some of its potential use cases,
optimization aspects, and challenges. Specifically, we argued that a carefully
designed enhanced control architecture is essential in order to take full
advantage of the RIS $3$D passive beamforming capabilities and the flexibility
of flying devices. In contrast to existing SoA frameworks, our envisioned
enhanced control architecture is able _to bridge together several
conventionally isolated entities and exploit useful information generated by
the UAV sensors_ (in addition to the user feedback on the perceived QoS) in
order to adapt the system configuration to varying environmental conditions.

We analyzed two practical scenarios of interest wherein the underlying A2G
network benefits from our proposed architecture and quantified its performance
in a relevant case study. Our results show that $i$) our proposed _Adaptive_
scheme is able to adapt the CSI and RIS beamforming strategy as required
according to varying UAV speeds, $ii$) the _adaptability_ comes at a non-
negligible reconfiguration overhead cost of about $5$ to $15$% and $iii$) the
proposed _adaptive_ solution successfully manages to keep the data rate at low
degradation percentages ($<10\%$) for the UAV speed range considered ($5$ to
$50$ km/h).

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## Biographies

Francesco Devoti (M’20) received the B.S., and M.S. degrees in
Telecommunication Engineering, and the Ph.D. degree in Information Technology
from the Politecnico di Milano, in 2013, 2016, and 2020 respectively. He is
currently a research scientist at NEC Laboratories Europe. His research
interests include millimeter-wave technologies in 5G networks.
---
Placido Mursia (S’18–M’21) received the B.Sc. and M.Sc. (with honors) degrees
in Telecommunication Engineering from Politecnico of Turin in 2015 and 2018,
respectively. He obtained his Ph.D from Sorbonne Université of Paris, at the
Communication Systems department of EURECOM in 2021. He is currently a
research scientist in the 5GN group at NEC Laboratories Europe. His research
interests lie in convex optimization, signal processing and wireless
communication.
---
Vincenzo Sciancalepore (S’11–M’15–SM’19) received his M.Sc. degree in
Telecommunications Engineering and Telematics Engineering in 2011 and 2012,
respectively, whereas in 2015, he received a double Ph.D. degree. Currently,
he is a Principal Researcher at NEC Laboratories Europe, focusing his activity
on reconfigurable intelligent surfaces. He is an Editor of the IEEE
Transactions on Wireless Communications.
---
Xavier Costa-Perez (M’06–SM’18) is Research Professor at ICREA, Scientific
Director at the i2Cat R&D Center and Head of 6G Networks R&D at NEC
Laboratories Europe. Xavier served on the Program Committee of several
conferences (including IEEE ICC and INFOCOM), published at top research venues
and holds several patents. He received his M.Sc. and Ph.D. degrees in
Telecommunications from the Polytechnic University of Catalonia (UPC) in
Barcelona.
---