# Little or no equalization is needed in energy-efficient sub-THz mobile
access

Lorenzo Miretti, Thomas Kühne, Alper Schultze, Wilhelm Keusgen, Giuseppe
Caire, Michael Peter, Sławomir Stańczak, Taro Eichler L. Miretti, A. Schultze,
M. Peter, and S. Stańczak are with the Fraunhofer Heinrich Hertz Institute
(email: {lorenzo.miretti, alper.schultze, michael.peter,
slawomir.stanczak}@hhi.fraunhofer.de). L. Miretti and S. Stańczak are also
with the Technische Universität Berlin. T. Kühne, W. Keusgen, and G. Caire are
with the Technische Universität Berlin (email: {thomas.kuehne,
wilhelm.keusgen, caire}@tu-berlin.de). T. Eichler is with Rohde & Schwarz,
Munich (email: taro.eichler@rohde-schwarz.com). The authors acknowledge the
financial support by the Federal Ministry of Education and Research of Germany
in the programme of “Souverän. Digital. Vernetzt.” Joint project 6G-RIC,
project identification numbers: 16KISK020K, 16KISK030.

###### Abstract

By trading coverage and hardware complexity for abundance of spectrum, sub-THz
mobile access networks are expected to operate under highly directive and
relatively spectrally inefficient transmission regimes, while still offering
enormous capacity gains over current sub-6GHz alternatives. Building on this
assumption, and supported by extensive indoor directional channel measurements
at $160\text{\,}\mathrm{GHz}$, this study advocates the use of very simple
modulation and equalization techniques for sub-THz mobile access.
Specifically, we demonstrate that, under the aforementioned transmission
regimes, little or no equalization is needed for scoring significant capacity
gain targets. In particular, we show that single-carrier or low-number-of-
subcarriers modulations are very attractive competitors to the dramatically
more complex and energy inefficient traditional multi-carrier designs.

###### Index Terms:

sub-THz, energy efficiency, single-carrier, waveform design, 6G.

††publicationid: pubid:  This work has been submitted to the IEEE for possible
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##  I Energy-efficient sub-THz mobile access

###  I-A Spectral efficiency is neither needed nor welcome

The exploitation of the large portions of available spectrum in the sub-THz
band (90-300 GHz) is one of the most promising directions for enhancing the
capacity of current mobile access networks [1, 2]. In contrast to current
sub-6GHz networks, for which a 100-fold capacity increase can only be achieved
by means of extreme spatial multiplexing and very complex modulation schemes,
sub-THz networks can score this ambitious goal by transmitting fewer
simultaneous data streams with relatively low spectral efficiency
($\approx$1-3 bit/s/Hz). These observations are best illustrated by focusing
on the following approximate formula for the network capacity:

$C=B\cdot M\cdot\mathrm{SE}\quad\text{bit}/\text{s}/\text{km}^{2},$ (1)

where $B$ is the signal bandwidth in Hz, $M$ is the number of spatially
multiplexed streams per $\text{km}^{2}$, and SE is the per-stream spectral
efficiency in bit/s/Hz. In excellent conditions, current 5G sub-6GHz networks
approximately support $B\approx 100$ MHz, $M\approx 16$, and
$\mathrm{SE}\approx 3-6$ bit/s/Hz, by using massive MIMO arrays with $64$
antennas and multi-carrier $256$-QAM modulated signals. A 100-fold capacity
increase at sub-6GHz frequencies would require a 100 times larger aggregate
spectral efficiency (i.e., the product of $M$ and $\mathrm{SE}$), which is
very challenging to achieve due to the problematic interaction between $M$ and
$\mathrm{SE}$ caused by interference, and since $\mathrm{SE}\approx 10$
bit/s/Hz is already close to its practical limit caused by hardware
imperfections. In contrast, by moving to sub-THz frequencies, for which
bandwidths up to $B\approx 30$ GHz are conceivable, the same goal could be
achieved with a dramatically more relaxed requirement on both $M$ and
$\mathrm{SE}$.

Admittedly, if the target $100$-fold capacity increase must be realized in an
energy efficient manner, finding the optimal trade-off between bandwidth and
spectral efficiency is a highly non-trivial task. For instance, the best known
analytic tools do not cover the energy consumed by the hardware [3]. The
present study focuses on sub-THz mobile access, i.e., on the exploration of
the very large bandwidth extreme of this trade-off. In this regime,
restricting to modulation schemes with low spectral efficiency is not only
sufficient but also of paramount importance, since the resulting relaxation of
the hardware requirements offers a unique opportunity for developing sub-THz
transceivers with tolerable energy consumption [2].

###  I-B Highly directive steerable beamforming antennas

To guarantee reasonable coverage at such high frequencies and large bandwidths
with acceptable radiated power, highly directive antennas must be used [1].
For instance, upgrading an ideal sub-6GHz free-space link through a $100$-fold
increase in frequency and bandwidth, while keeping the same coverage, radiated
power, and target spectral efficiency, would require a directivity gain of
about $60$ dB. For this reason, even by considering significantly more relaxed
coverage and spectral efficiency requirements than their sub-6GHz
counterparts, sub-THz mobile access networks will likely need directive
antennas ($\approx 10-30$ dBi gain) at both the transmit and receive ends.
Furthermore, due to user mobility, high directivity must be dynamically
realized via steerable beamforming antennas. This poses a series of remarkable
technical challenges subject of extensive ongoing research, such as the energy
efficient design of such type of antennas, and the so-called two-sided beam
alignment problem [4, 5]. The present study assumes that high-gain steerable
beamforming antennas are indeed feasible, and that a beam alignment procedure
can be successfully performed. Under this assumption, the main focus is the
evaluation of the possibly beneficial impact of using highly directive
antennas on channel equalization.

###  I-C The impact of directivity on inter-symbol interference

(a)

(b)

Figure 1: Pictorial representation of the radio channel comprising of a
multipath propagation channel and (a) low directivity antennas; (b) high
directivity antennas. In contrast to case (a), using highly directive antennas
as in case (b) significantly attenuates most multipath components of the
propagation channel. As shown experimentally in the remainder of this study,
this effect reduces inter-symbol interference, and hence simplifies
equalization.

Many previous studies on millimiter-wave and THz systems already recognized
that the use of highly directive antennas leads to radio channels (i.e.,
comprising the effect of the antennas) with considerably lower delay spread
than the underlying propagation channels, and hence to radio channels for
which simple modulation and equalization techniques are potentially sufficient
[6, 7, 8]. This crucial observation can be explained by the so-called spatial
filtering effect, which is illustrated in Figure 1. As customary, we model the
single-input single-output radio channel between two beamforming antennas by
means of a channel impulse response $h=(h_{0},\ldots,h_{L-1})$ with $L$ taps,
and a nominal SNR parameter. Then, by assuming standard coding techniques, and
by treating inter-symbol interference as noise, an achievable spectral
efficiency over this channel is

$\mathrm{SE}=\log_{2}\left(1+\dfrac{|h_{l_{0}}|^{2}}{\sum_{l\neq
l_{0}}|h_{l}|^{2}+\mathrm{SNR}^{-1}}\right),$ (2)

where $l_{0}$ is the delay corresponding to the main peak of the power delay
profile. Importantly, we remark that the above transmission scheme does not
consider any form of digital equalization. For this reason, in the presence of
significant inter-symbol interference, as depicted in Figure 1a for the case
of low antenna directivity, the spectral efficiency may be quite poor.
However, when highly directive antennas are employed and most multipath
components are consequently attenuated, as depicted in Figure 1b, this simple
scheme may be sufficient to achieve the relatively low target spectral
efficiencies of sub-THz mobile access networks. Moreover, if needed, simple
equalization techniques may be sufficient to suppress most of the residual
inter-symbol interference and further improve upon (2). The reminder of this
study is devoted to a deeper investigation of this intuition.

###  I-D The impact of directivity on multi-user interference

The spectral efficiency expression (2) based on treating inter-symbol
interference as noise can be readily extended such that additional
interference terms corresponding to the incoming signal of other users are
included. Following the same spatial filtering argument as for the inter-
symbol interference, the use of highly directive antennas may significantly
mitigate the impact of multi-user interference, provided that the interfering
signal components do not overlap with the main desired signal component. In
this work, for simplicity, we neglect multi-user interference by invoking the
aforementioned spatial filtering effect, and by assuming the concurrent
scheduling of users that are sufficiently separated in the angular domain.
Therefore, the resulting capacity figures are only valid up to some maximum
multiplexing gain $M$ that, roughly speaking, depends on the antenna beamwidth
and the angular distribution of the users.

###  I-E Objective, methodology, and organization of the paper

Following the above discussion, the main objective of this study is to
validate experimentally the claim that the combination of a relatively low
target per-stream spectral efficiency and the use of highly directive
beamforming antennas may allow the use of simpler modulation and equalization
techniques than in traditional designs, and hence leave room for significant
energy savings. To this end, differently than in most related studies (see,
e.g., [9, 10]), we follow an unconventional yet more targeted methodology that
integrates accurate measurement data with rigorous information theoretic
arguments. More specifically, instead of extracting channel modelling
parameters such as delay spread, we present the results of a measurement
campaign directly in terms of theoretically achievable performance, for
different choices of modulation and equalization techniques. The reminder of
this study is organized as follows: Section II illustrates the performed
measurement campaign, and discusses how the obtained measurements can be used
to approximate realistic sub-THz communication channels. Section III presents
the results of the measurement campaign in terms of achievable spectral
efficiency and coded bit error rate (BER) for promising modulation and
equalization techniques. Finally, Section IV summarizes the main conclusions
and outlines some possible future directions.

##  II Measurement campaign

This section illustrates the technical details of a measurement campaign
aiming at characterizing the performance of simple modulation and equalization
techniques for realistic indoor radio channels with $160\text{\,}\mathrm{GHz}$
carrier frequency, $4\text{\,}\mathrm{GHz}$ bandwidth, and directive
beamforming antennas.

###  II-A Channel sounder setup

The adopted channel sounder is based on the principle of time-domain channel
measurement. Test and measurement equipment working in the millimeter-wave
frequency range is expanded with active D-band front-ends that allow extension
to carrier frequencies between $110\text{\,}\mathrm{GHz}$ and
$170\text{\,}\mathrm{GHz}$. The setup can be divided into transmitter (TX) and
receiver (RX), where the TX is kept fixed and the RX is moved across different
positions. The TX consists of a signal generator (R&S® SMW200A) and an
external D-Band front-end (R&S® FE170ST). The signal generator provides a
perfect periodic correlation sequence (Frank-Zadoff-Chu sequence) with a
measurement bandwidth of $4\text{\,}\mathrm{GHz}$ and a sequence duration of
$50\text{\,}\mathrm{\SIUnitSymbolMicro s}$ at an intermediate frequency. The
external front-end then up-converts the provided signal to the desired carrier
frequency of $160\text{\,}\mathrm{GHz}$. As transmitting antenna, a standard
gain horn with a gain of $25\text{\,}\mathrm{dBi}$ is used. The RX consists of
a signal analyzer (R&S® FSW43) and another external D-Band front-end (R&S®
FE170SR). The external front-end on the RX down-converts the received signal
to an intermediate frequency. The signal analyzer samples the received signal
and stores the IQ samples. The synchronization between transmitter and
receiver is achieved by using a reference signal of $10\text{\,}\mathrm{MHz}$.
The receiver is triggered synchronously to the sequence start by a versatile
pulse source (Synchronomat). Hence, coherent sampling and the measurement of
time-of-flight are ensured. An open waveguide with $6\text{\,}\mathrm{dBi}$
gain serves the receiving antenna. The chosen antennas emulate a typical sub-
THz system with a large steerable antenna at the access point and a smaller
steerable antenna at the user equipment after beam alignment.

###  II-B Measurement scenario and procedure

The measurement campaign is conducted on the company premises of Rohde &
Schwarz in Munich, Germany. As venue, the rectangular atrium of a large
building is used. The atrium’s dimensions are $15\text{\,}\mathrm{m}$ x
$50\text{\,}\mathrm{m}$ with a height of approximately
$20\text{\,}\mathrm{m}$. The environment is mainly characterized by glass
walls, a tiled floor, metallic surfaces (elevators) and concrete pillars.
Figure 2 illustrates the measurement setup in terms of a floor plan, and the
chosen TX and RX positions. For all positions, a line-of-sight (LoS) between
TX and RX was ensured, and the TX and RX antennas were aligned along the
corresponding LoS direction. This corresponds to the assumed use case where
the beams of the steerable antennas at the TX and the RX are aligned. Two
different measurement scenarios were considered, differing in the movement
pattern within the atrium. In the first scenario, the TX was placed centrally
at the beginning of the atrium, and the RX was moved longitudinally. Distances
from $1\text{\,}\mathrm{m}$ up to $40\text{\,}\mathrm{m}$ were considered,
resulting in $20$ measurement positions. In the second scenario, a similar
movement pattern was used along the diagonal line connecting two opposite
corners of the atrium. Distances from $1\text{\,}\mathrm{m}$ to
$46\text{\,}\mathrm{m}$ were considered, resulting in $26$ measurement
positions.

The first scenario was designed such that multipath components may be
detectable, including ground reflections and reflections by, e.g., the metal
surfaces of the elevators. The second scenario was designed such that fewer
multipath components should be detectable with respect to the first scenario.
Nevertheless, the ground reflection may still be detectable.

The collected raw data are processed as follows: First, for each measurement
position and scenario, a set of $500$ calibrated channel impulse responses are
extracted from the IQ samples, each corresponding to a single sounding
sequence. Across the $500$ measurements a phase deviation is estimated under
the assumption of a stationary radio channel. Thus, the common phase error of
each sequence is compensated and the $500$ complex impulse responses are
averaged, resulting in one common impulse response per measurement position
with improved correlation gain. Finally, the obtained impulse responses are
power normalized, and synchronized with respect to the the first peak.
Accurate synchronization with sub-sample resolution is achieved via upsampling
and cubic interpolation of the main peak. The resulting channel impulse
responses correspond to channels as seen by a hypothetical communication
system after phase and frequency offset compensation and time synchronization.
Furthermore, since the impulse responses show a dynamic range of more than
$60\text{\,}\mathrm{dB}$ they can be considered as noise free channel
estimates for all SNR regions of practical communication systems.

Figure 2: Schematic visualization of the measurement scenario. The upper
drawing is a top-down illustration of the atrium’s dimension and composition.
The green star represents the TX’s first position and the blue star it’s
second. The dashed lines indicate both the TX’s as well the RX’s orientation
during the measurement campaign’s execution. The lower drawing illustrates the
TX’s and RX’s height, as well as the measured distances.

###  II-C Illustrative examples of measured channels

Figure 3: Power delay profile of three distinguishing measurements (red, green
and blue) with enlarged range around the first peak (upper right corner). The
time axis is given in channel taps as integers (one tap corresponds to
$0.25\text{\,}\mathrm{ns}$).

The exceptionally high dynamic range allows the observation of multipath
components that are significantly smaller than the LoS path. Additionally, the
temporal resolution resulting from the measurement bandwidth of
$4\text{\,}\mathrm{GHz}$ allows the distinction of paths with a theoretical
minimum distance of $0.25\text{\,}\mathrm{ns}$. Figure 3 shows the measured
power delay profile $|h|^{2}$ for three representative cases. The green plot
refers to a measurement from the first scenario, at the distance of
$18\text{\,}\mathrm{m}$. Importantly, the green plot reveals the presence of a
strong multipath component only $2$ channel taps away from the LoS path, and
it is a representative example of a channel with significant inter-symbol
interference. This path is likely due to a ground reflection. The red plot and
the blue plot refer to two measurements from the second scenario, at a
distance of of $7\text{\,}\mathrm{m}$ and $1\text{\,}\mathrm{m}$,
respectively. In the red plot, the LoS path is clearly separable from the
other paths, which all lie well below $-30\text{\,}\mathrm{dB}$. This is a
representative example of a particularly well behaved channel with negligible
inter-symbol interference. In the blue plot, two relatively strong multipath
components can be observed within a range of $0\text{\,}\mathrm{dB}$ and
$-30\text{\,}\mathrm{dB}$, at a delay of $2$ and $30$ taps, respectively. Both
components, and especially the second path, may be caused by the interaction
with the metal surfaces of the TX and RX devices themselves. This is a
representative example of a channel with several significant multipath
components with intermediate strengths that might influence the performance of
a communication system. Due to the high dynamic range of the measurement
setup, more multipath components are visible in all three channels below
$-40\text{\,}\mathrm{dB}$. Nonetheless, these components are so weak that they
can be neglected for the low spectral efficiency regime.

Overall, all measured radio channels appear to be composed only by few
relevant propagation paths with significant strength. These propagation paths,
corresponding to the peaks of the power delay profiles, are mostly due to
reflections captured by the main lobes of the antenna radiation patterns, and
are generally located within $10-30$ taps. This confirms the main intuition
illustrated in Figure 1. However, differently than in the idealized model of
Figure 1, the channels exhibit additional taps in the close proximity of the
channel peaks, which are likely due to the chosen pulse shape. In fact, the
measurements are hard band-limited, i.e., they consider a sinc-like Nyquist
pulse with pronounced side lobes, which contribute to inter-symbol
interference in the (almost sure) case of the path delay being not perfectly
aligned with the sampling grid.

###  II-D Relation with realistic sub-THz communication channels

As already anticipated, in the reminder of this study we assess the
performance of simple modulation and equalization techniques directly over the
measured channel impulse responses. A natural question is: to what extend
these impulse responses reflect the channel seen by a realistic communication
system? First, we point out that the chosen antenna directivities ($25$ dBi
and $6$ dBi) can be reasonably implemented using steerable phased arrays of
practical size and cost, and, as discussed in Section I, are almost necessary
to provide sufficient coverage at sub-THz frequencies. Note that, since beam
alignment is implemented by means of mechanical steering (a form of true-time-
delay beamforming), our measurements are insensitive to the spatial wideband
effect [11] for the LoS path. Nevertheless, the chosen antenna directivities
and signal bandwidth fall into a regime where the spatial wideband effect is
negligible for $\lambda/2$ spaced phased arrays, and therefore the
measurements are also representative of more practical beamforming
implementations. Second, we argue that the chosen pulse shape still produces
good approximations of realistic communication channels. In fact, although
practical communication systems may consider pulses with faster decay, such as
a root raised cosine pulses, the difference would be mostly noticeable for the
weaker secondary paths, since accurate time recovery is performed with respect
to the main peak. Finally, we remark that the current channel impulse
responses are representative of a system with $4$ GHz communication bandwidth,
which is a reasonable choice for early prototypes of sub-THz mobile access
networks. Nevertheless, this bandwidth is already sufficient to capture the
main impact of the absolute delay spread with a sub-ns resolution. Therefore,
the results in this study could also be approximately extended to systems with
higher bandwidths, provided that the delay spread expressed in number of
channel taps is scaled accordingly.

##  III Measurement results

In this section we present the results of our measurement campaign in terms of
achievable performance of promising modulation and equalization techniques
over the measured channels. In particular, motivated by the importance of
energy-efficient designs, we study two alternatives to traditional
implementations of multi-carrier modulation based on orthogonal frequency-
division multiplexing (OFDM). Multi-carrier modulation is known to approach
the fundamental spectral efficiency limits of inter-symbol interference
channels with a relatively simple equalization effort, provided that the
number of subcarriers is large enough. However, its main drawback is the known
issue of producing signals with peak-to-average power ratio (PAPR)
proportional to the number of subcarriers, that needs to be compensated by
significantly reducing the energy efficiency of the transceiver front-end.
Therefore, we consider single-carrier modulation with linear equalization and
multi-carrier modulation with a low number of subcarriers. We then show that
both alternatives achieve spectral efficiencies of about $1-3$ bit/s/Hz over
all measured channels. Hence, both schemes may be sufficient to score the
target capacity gains in a more energy efficient manner.

Figure 4: Achievable spectral efficiency for single-carrier modulation with
$N$-taps linear equalization, in the considered indoor scenario assuming
$\mathrm{SNR}=6$ dB. The line plots are the median value over all measurement
points, and the matched filter upper bound. The variations across measurement
points are captured by the empirical cumulative density function, illustrated
through a color map, and truncated within the range of observed spectral
efficiencies. Due to the specific structure of the measured channels, single-
carrier modulation with very few equalization taps appears sufficient to
achieve spectral efficiencies of about $1.4-2.3$ bit/s/Hz, and, hence, score
the capacity gains promised by sub-THz networks in an energy efficient manner.

###  III-A Single-carrier with linear equalization

We first consider single-carrier modulation followed by a $N$-tap linear
equalizer. An achievable spectral efficiency is given by (2), with $h$
replaced by its convolution with the equalizer impulse response. We compute
the optimal equalizer coefficients in terms of spectral efficiency, i.e.,
maximizing the corresponding Rayleigh quotient appearing in (2). The solution
takes the form of a standard linear minimum mean-square error equalizer. We
set $l_{0}$ as the delay of the main peak of the original power delay profile,
plus an additional offset which is tuned such that the equalizer focuses on
the optimal number of precursors and postcursors (in practice, only one or two
postcursors are relevant).

Figure 4 reports the resulting achievable spectral efficiency over the
measured channels, as a function of the number of taps $N$, where $N=1$ stands
for no equalization, and by assuming $\mathrm{SNR}=6$ dB. Since the channel
impulse responses are normalized, the SNR parameter can be interpreted as a
receive SNR comprising of transmit power, path loss, beamforming gain, and
receiver noise. The dashed line shows the so-called matched filter upper
bound, equivalent to the capacity of an AWGN channel with the same SNR, which
provides an upper bound on the achievable spectral efficiency. We remark that
this bound is generally not achievable and may be even quite far from the
achievable performance under ideal equalization, in channels with significant
inter-symbol interference. The chosen SNR is in the high range of what is
expected to be practically feasible at the edge of a sub-THz mobile access
network. Furthermore, it leads to an upper bound within the spectral
efficiency regime of interest, i.e., $1-3$ b/s/Hz. For lower SNR values, the
gains of the equalization decrease until they become negligible. A higher SNR
improves performance for all choices of number of taps $N$, up to saturation
points driven by the uncompensated inter-symbol interference for the given
choice of $N$.

A we can see, $50\%$ of the measured channels are so well-behaved that no
equalization is actually needed to approach almost ideal performance.
Furthermore, reliable communication above $1$ bit/s/Hz without equalization is
theoretically possible in the vast majority of the cases. However, in the
other $50\%$ of the cases, the inter-symbol interference caused by strong
paths in the close vicinity of the main peak causes large variations from the
ideal performance. In a few extreme cases the achievable spectral efficiency
may even drop below $1$ bit/s/Hz. Fortunately, this residual inter-symbol
interference can be significantly mitigated by using only $5-6$ equalization
taps, for which the achievable spectral efficiency is consistently kept above
$1.4$ bit/s/Hz. One important comment here is that, in all measured channels,
further spectral efficiency gains can only be observed after adding several
tens or even hundreds of equalization taps, i.e., when the filter length
reaches the weaker multipath components located far away from the main peak.
Nevertheless, these gains are negligible and do not justify the longer
equalizers.

As a final remark, we recall that the above figures are valid for a $4$ GHz
communication bandwidth, i.e., for theoretical rates on the order of $4$
Gbit/s/stream. If higher bandwidths are used, and the target SNR is kept to
the same value, the number of required equalization taps for each measurement
point may increase proportionally, reaching few tens of taps in the worst
cases. Still, the main message of this section would continue holding, i.e.,
single-carrier modulation and a simple linear equalizer may be sufficient to
score the target capacity goals in the considered indoor scenario.

###  III-B Multi-carrier with a low number of subcarriers

Figure 5:  Achievable spectral efficiency for multi-carrier modulation with
$K$ subcarriers and no prefix, in the considered indoor scenario assuming
$\mathrm{SNR}=6$ dB. The line plots are the median value over all measurement
points, and the matched filter upper bound. The variations across measurement
points are captured by the empirical cumulative density function, illustrated
through a color map, and truncated within the range of observed spectral
efficiencies. Using a low number of subcarriers and no prefix is sufficient to
score the target capacity goals, with a much better PAPR than conventional
OFDM.

In this section we consider multi-carrier modulation based on a standard OFDM
implementation with $K$ subcarriers and a cyclic prefix of length $Q$. In
contrast to traditional designs, we study the effect of choosing a prefix
length much shorter than the delay spread, i.e., we focus on $Q\ll L$. In this
regime, the channel in the subcarrier domain suffers from both inter-carrier
interference (ICI) and inter-block interference (IBI) [12]. By following
similar arguments as for the single-carrier case, i.e., by treating ICI and
IBI as noise, an achievable spectral efficiency is given by the sum of $K$
expressions similar to (2), where the signal and inter-symbol interference
terms are replaced by the appropriate channel coefficients in the subcarrier
domain, and a pre-log factor which takes into account the prefix overhead. For
all tested numbers of subcarriers $K$, we optimize the prefix length $Q$.
Furthermore, similarly to the single-carrier case, we consider a block-
boundary detection stage such that the prefix is optimally centered with
respect to the main channel peak.

Interestingly, for all tested numbers of subcarriers, the gains of adding a
cyclic prefix in terms of ICI and IBI mitigation are minor and outweighed by
the multiplicative spectral efficiency losses due to the overhead. As a
consequence, we focus on an unconventional implementation with no prefix
($Q=0$), as proposed in [12]. However, differently than [12], we do not
consider additional equalization stages, and simply treat ICI and IBI as
noise. Similar to the single-carrier case, Figure 5 reports the resulting
achievable spectral efficiency over the measured channels as a function of the
number of subcarriers $K$ and by assuming $\mathrm{SNR}=6$ dB. We notice that
very few subcarriers are needed to achieve spectral efficiencies significantly
higher than $1$ bit/s/Hz in all measured channels and are often sufficient to
be close to the ideal performance limit. An informal explanation is that, even
without a cyclic prefix, the inter-symbol interference is spread over multiple
subcarriers and pushed towards the noise floor as $K$ increases. Furthermore,
at the price of a higher PAPR and hence of more stringent hardware efficiency
requirements, we observe that the proposed multi-carrier modulation for $K\geq
16$ achieves better spectral efficiencies (in both median and worst case) than
single-carrier modulation even for a large number of equalization taps (we
tested up to $N=256$ equalization taps).

###  III-C Performance of a complete system

Figure 6: BER versus SNR for single-carrier QPSK modulated signals and
$7$-taps linear equalization, in the considered indoor scenario. The median,
worst, and best performance over all measured points are shown. The coded BER
curves refer to standard LDPC codes with rate 1/2 and block length $1296$, and
show that reliable communication at $1$ bit/s/Hz is indeed feasible by means
of simple and energy-efficient modulation and equalization techniques.

In this section, we show that reliable communication at
$1\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{H}\mathrm{z}$
over the measured channels is indeed feasible by using simple modulation and
equalization techniques. Specifically, we run a numerical simulation of a
system employing single-carrier QPSK modulated signals, linear minimum mean-
square error equalization with $N=7$ taps (two of which are postcursors), and
standard low-density parity-check (LDPC) codes. The LDPC code has a rate of
$1/2$, a $1296$ bit block length, and uses the parity-check matrix as defined
in the IEEE 802.11n standard. The LDPC decoder is tuned by treating inter-
symbol interference as additive Gaussian noise with the same power and uses a
belief propagation algorithm. The simulated transmit signals are distorted by
the measured channel impulse responses and Gaussian noise is added according
to the SNR parameter. Figure 6 reports the performance in terms of coded BER
for all measured channels as a function of the SNR. For reference, we also
report the corresponding uncoded BER curves, and a lower bound on the minimum
SNR for reliable communication at $1$ b/s/Hz, obtained from the aforementioned
matched filter upper bound on the achievable spectral efficiency (Shannon
limit). Remarkably, the gap between the obtained BER curves and the (not
necessarily achievable) Shannon limit is in the vast majority of cases between
$2.5-4$ dB, a range which is in line with the typical gap from capacity of
LDPC codes on AWGN channels. Furthermore, letting $\mathrm{SNR}\approx 6$ dB
allows for reliable communication even for the worst-case channels, as
predicted by Figure 4. This confirms that, for low target spectral
efficiencies, inter-symbol interference can be indeed successfully mitigated
by means of simple equalization techniques, at the point that it can be
treated as noise and further mitigated by standard codes for the AWGN channel.

##  IV Conclusion

Our results corroborate the attractiveness of simple and energy efficient
alternatives to traditional OFDM-based designs in sub-THz mobile access
networks. In particular, our results strengthen the claim that single-carrier
modulation is a promising solution for the early development of these
networks, where energy efficiency is a major limiting factor. Then, our
results suggest that an evolution towards more capable networks can be
accomplished by gradually increasing the number of subcarriers in an
unconventional OFDM implementation. Our methodology integrates accurate
experimental data with rigorous information theoretic arguments. The
integration of these approaches is particularly effective, since our channel
measurements closely approximate the impulse response as seen by a
hypothetical yet realistic sub-THz communication system in the same indoor
environment, and with the same signal bandwidth ($4$ GHz), carrier frequency
($160$ GHz), and antenna directivities at the access point ($25$ dBi) and user
equipments ($6$ dBi). To further test the robustness of our analysis, future
works should explore different system parameters. For instance, even though
some preliminary conclusions could already be extrapolated from this study, a
direct performance evaluation for larger bandwidths is desirable. Perhaps most
importantly, future works should also cover different environments with more
reflections and possibly a higher number of multipath components. Furthermore,
non-line-of-sight propagation and the effect of multi-user interference are
interesting additions to the analysis.

## References

  * [1] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. C. Trichopoulos,  “Wireless communications and applications above 100 GHz: Opportunities and challenges for 6G and beyond,”  IEEE Access, vol. 7, pp. 78729–78757, 2019.
  * [2] N. Rajatheva, I. Atzeni, S. Bicais, E. Björnson, A. Bourdoux, S. Buzzi, C. D’Andrea, J.-B. Dore, S. Erkucuk, M. Fuentes, et al.,  “Scoring the terabit/s goal: Broadband connectivity in 6G,”  arXiv preprint arXiv:2008.07220, 2020.
  * [3] S. Verdú,  “Spectral efficiency in the wideband regime,”  IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1319–1343, 2002\.
  * [4] H. Yan, S. Ramesh, T. Gallagher, C. Ling, and D. Cabric,  “Performance, power, and area design trade-offs in millimeter-wave transmitter beamforming architectures,”  IEEE Circuits Syst. Mag., vol. 19, no. 2, pp. 33–58, 2019.
  * [5] X. Song, T. Kühne, and G. Caire,  “Fully-/partially-connected hybrid beamforming architectures for mmwave MU-MIMO,”  IEEE Trans. Wireless Commun., vol. 19, no. 3, pp. 1754–1769, 2020\.
  * [6] R. Sun, C. A. Gentile, J. Senic, P. Vouras, P. B. Papazian, N. T. Golmie, and K. A. Remley,  “Millimeter-wave radio channels vs. synthetic beamwidth,”  IEEE Commun. Mag., vol. 56, no. 12, pp. 53–59, 2018.
  * [7] R. Piesiewicz, M. Jacob, M. Koch, J. Schoebel, and T. Kürner,  “Performance analysis of future multigigabit wireless communication systems at THz frequencies with highly directive antennas in realistic indoor environments,”  IEEE J. Select. Topics Quantum Electron., vol. 14, no. 2, pp. 421–430, 2008.
  * [8] H. Sarieddeen, M.-S. Alouini, and T. Y. Al-Naffouri,  “An overview of signal processing techniques for terahertz communications,”  Proc. IEEE, vol. 109, no. 10, pp. 1628–1665, 2021.
  * [9] J. Gomez-Ponce, N. A. Abbasi, A. E. Willner, C. J. Zhang, and A. F. Molisch,  “Directionally resolved measurement and modeling of THz band propagation channels,”  IEEE Open Journal of Antennas and Propagation, vol. 3, pp. 663–686, 2022.
  * [10] Y. Xing, T. S. Rappaport, and A. Ghosh,  “Millimeter wave and sub-THz indoor radio propagation channel measurements, models, and comparisons in an office environment,”  IEEE Commun. Lett., vol. 25, no. 10, pp. 3151–3155, 2021.
  * [11] B. Wang, F. Gao, S. Jin, H. Lin, G. Y. Li, S. Sun, and T. S. Rappaport,  “Spatial-wideband effect in massive MIMO with application in mmWave systems,”  IEEE Commun. Mag., vol. 56, no. 12, pp. 134–141, 2018.
  * [12] M. Toeltsch and A. F. Molisch,  “Efficient OFDM transmission without cyclic prefix over frequency-selective channels,”  in Proc. 11th IEEE Int. Symp. Personal Indoor and Mobile Radio Commun., 2000, vol. 2, pp. 1363–1367.

## Biographies

Lorenzo Miretti received the B.Sc. and M.Sc. degrees in Telecommunication
Engineering from Politecnico di Torino in 2015 and 2018, respectively, and the
Ph.D. degree in wireless communications from EURECOM and Sorbonne Université
in 2021. He is currently a post-doctoral researcher with the Technical
University of Berlin and the Fraunhofer Heinrich Hertz Institute. He
investigates novel solutions for next generation wireless networks, such as
cell-free massive MIMO and sub-THz mobile access networks.

Thomas Kühne received the Dipl.-Ing. (equivalent to an M.Sc.) degree in
Electrical Engineering from the University of Technology Dresden, Germany, and
the Dr. -Ing. (Ph.D.) degree in 2022 in wireless communication from the
Technische Universität Berlin, Germany. During his master’s studies, he
focused on communication systems and circuit design, while during his Ph.D.
studies, he specialized in wireless communication systems and signal
processing. Since 2015, he has been working in Prof. Caire’s Communication and
Information Theory group at Technische Universität Berlin, currently as a
Postdoc researcher. His research interests include the co-design between the
hardware, the signal processing, and the system design of next generation
wireless communication systems.

Alper Schultze received the B.Sc. degree and the M.Sc. degree in electrical
engineering from the Technical University of Berlin, Germany, in 2016 and
2019, respectively. During his studies, he specialized on communication
systems and spent one year abroad in Italy at the University of Bologna
deepening his knowledge on telecommunication engineering. In 2019, he joined
the Fraunhofer Heinrich Hertz Institute in Berlin, as a research associate.
His research focuses on channel measurement and characterization in the sub-
THz and THz domain.

Wilhelm Keusgen received the Dipl.-Ing. (M.S.E.E.) and Dr.-Ing. (Ph.D.E.E.)
degrees from the RWTH Aachen University, Aachen, Germany, in 1999 and 2005,
respectively. From 1999 to 2004, he was with the Institute of High Frequency
Technology, RWTH Aachen University, where he worked on the reciprocity of
multiple antenna systems. From 2004 to 2021 he was heading a research group
for millimeter-waves and advanced transceiver technologies at the Fraunhofer
Institute for Telecommunications, Heinrich Hertz Institute, located in Berlin,
Germany. Since 2021 he is a full professor at Technische Universität Berlin
and heading the chair for Microwave Systems. His main research areas are
millimeter wave and THz communications for 5G and beyond.

Giuseppe Caire received his B.Sc. degree from Politecnico di Torino, Italy, in
1990, his M.Sc. degree from Princeton University, New Jersey, in 1992, both in
electrical engineering, and his Ph.D. degree from Politecnico di Torino in
1994. He has been a post-doctoral research fellow with the European Space
Agency (ESTEC, Noordwijk, The Netherlands) in 1994-1995, Assistant Professor
in Telecommunications at the Politecnico di Torino, Associate Professor at the
University of Parma, Italy, Professor with the Department of Mobile
Communications at the Eurecom Institute, Sophia-Antipolis, France, a Professor
of Electrical Engineering with the Viterbi School of Engineering, University
of Southern California, Los Angeles, and he is currently an Alexander von
Humboldt Professor with at the Technische Universität Berlin, Germany. Among
others, he received an ERC Advanced Grant in 2018, the IEEE Communications
Society Edwin Howard Armstrong Achievement Award in 2020, and the 2021 Leibniz
Prize of the DFG.

Michael Peter received the Dipl.-Ing. degree (M.S.) in electrical engineering
and information technology from the University of Karlsruhe, Germany, in 2004,
and the Dr.-Ing (Ph.D.) degree in electrical engineering, telecommunications,
from the Technische Universität Berlin, Germany, in 2018. He joined the
Fraunhofer Institute for Telecommunications, Heinrich Hertz Institute, Berlin,
Germany in 2005. In 2019, he became co-leader of the Millimeter Wave Research
Group and took over as head in 2021. His research interests include
millimeter-wave and THz communications with focus on channel measurements and
modeling, physical layer design and simulation, and performance analysis
taking into account hardware impairments. He is author and co-author of more
than 70 peer-reviewed scientific papers.

Sławomir Stańczak is Professor of Network Information Theory at the Technical
University of Berlin and Head of the Wireless Communications and Networks
Department at the Fraunhofer Heinrich Hertz Institute (HHI). Prof. Stańczak is
co-author of two books and more than 200 peer-reviewed journal articles and
conference papers in the field of information theory, wireless communications,
signal processing, and machine learning. Prof. Stańczak received research
grants from the German Research Foundation and the Best Paper Award from the
German Society for Telecommunications in 2014. He was an associate editor of
the IEEE Transactions on Signal Processing from 2012 to 2015 and chair of the
ITU-T Focus Group on Machine Learning for Future Networks including 5G from
2017 to 2020. Since 2020 Prof. Stańczak is chairman of the 5G Berlin
association and since 2021 he is coordinator of the projects 6G-RIC (Research
& Innovation Cluster) and CampusOS.

Taro Eichler is Technology Manager for wireless communications and photonics
at Rohde & Schwarz in Munich with a focus on 5G/6G technologies. Prior to
joining Rohde & Schwarz, Taro worked for Intel Corporation as specialist for
photonics communication solutions and for NTT Basic Research Laboratories. He
also worked on research projects at The University of Tokyo in the field of
quantum optics. He holds a diploma in physics from the Technical University of
Munich with thesis at the Max-Planck-Institute for Quantum Optics and a PhD in
physics from the University of Bonn, Germany.


# Charm CP violation and searches

Stefan Schacht  Department of Physics and Astronomy, University of Manchester,
Manchester M13 9PL, United Kingdom

###### Abstract

Charm CP violation is a unique gate to the physics of up-type quarks and
allows searches for physics beyond the Standard Model in new and exciting
ways, complementary to kaon and $b$ decays. We review recent advances,
focusing on symmetry-based methods and providing an outlook on the next
challenges at the intensity frontier of charm physics.

##  1 Introduction

Charm CP violation is a unique gate to explore the flavor structure of up-type
quarks and probe for physics beyond the Standard Model (BSM) in this sector.
In 2019 LHCb discovered charm CP violation by determining the difference of CP
asymmetries [1, 2]

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow
K^{+}K^{-})-a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow\pi^{+}\pi^{-})$
$\displaystyle=(-0.161\pm 0.028)\%\,,$ (1)

where

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow f)$
$\displaystyle\equiv\frac{|A(D^{0}\rightarrow
f)|^{2}-|A(\bar{D}^{0}\rightarrow f)|^{2}}{|A(D^{0}\rightarrow
f)|^{2}+|A(\bar{D}^{0}\rightarrow f)|^{2}}\,.$ (2)

The interpretation of Eq. (1) and the question if it implies BSM physics or
not is an on-going challenge for the theory community. Direct CP violation is
an interference effect of two amplitudes that have a relative weak and strong
phase. In terms of the underlying theory parameters of the two interfering
amplitudes, for charm decays we have [3]

$\displaystyle a_{CP}^{\mathrm{dir}}\approx
2r_{\mathrm{CKM}}r_{\mathrm{QCD}}\sin\varphi_{\mathrm{CKM}}\sin\delta_{\mathrm{QCD}}\,.$
(3)

Here, $r_{\mathrm{CKM}}$ is the magnitude of the ratio of the involved
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements of the two amplitudes,
$r_{\mathrm{QCD}}$ is the magnitude of the ratio of the relevant hadronic
matrix elements and $\varphi_{\mathrm{CKM}}$ and $\delta_{\mathrm{QCD}}$ are
the corresponding relative weak and strong phases, respectively.

In the Standard Model (SM), the interference effect resulting in
$a_{CP}^{\mathrm{dir}}\neq 0$ stems from the different ways one can reach the
final state of for example two pions from the initial state of a $D^{0}$. This
can proceed either directly via the CKM matrix elements $V_{cd}^{*}V_{ud}$, or
by first decaying to an intermediate state that contains strangeness, for
example $K^{+}K^{-}$, through the coupling $V_{cs}^{*}V_{us}$, which then
annihilates and forms a $\pi^{+}\pi^{-}$ state:

$\displaystyle D^{0}$
$\displaystyle\overset{V_{cd}^{*}V_{ud}}{\longrightarrow}\pi^{+}\pi^{-}\,,$
(4) $\displaystyle D^{0}$
$\displaystyle\overset{V_{cs}^{*}V_{us}}{\longrightarrow}K^{+}K^{-},\dots\overset{\mathrm{QCD}}{\longrightarrow}\pi^{+}\pi^{-}\,.$
(5)

The same happens completely analogously also for $K^{+}K^{-}$ final states:

$\displaystyle D^{0}$
$\displaystyle\overset{V_{cd}^{*}V_{ud}}{\longrightarrow}\pi^{+}\pi^{-},\dots\overset{\mathrm{QCD}}{\longrightarrow}K^{+}K^{-}\,,$
(6) $\displaystyle D^{0}$
$\displaystyle\overset{V_{cs}^{*}V_{us}}{\longrightarrow}K^{+}K^{-}\,.$ (7)

The interfering amplitudes have therefore a relative weak phase between the
CKM matrix element combinations $V_{cd}^{*}V_{ud}$ and $V_{cs}^{*}V_{us}$ as
well as a relative strong phase from the involved non-perturbative QCD
effects.

The fact that we know the involved CKM matrix elements from the global CKM
fit,  _i.e._ , from kaon and $b$ decays, leads to the SM estimate

$\displaystyle\Delta a_{CP}^{\mathrm{dir}}\sim 10^{-3}\times
r_{\mathrm{QCD}}\sin\delta_{\mathrm{QCD}}\,.$ (8)

However, we are as of now not able to reliably calculate the underlying
hadronic physics from first principles, so we do not know _a priori_ the size
of $r_{\mathrm{QCD}}$ and $\delta_{\mathrm{QCD}}$. The ratio
$r_{\mathrm{QCD}}$ is also known as the “penguin over tree” ratio. Note
however, that this commonly used term denotes more precisely just the ratio of
CKM-subleading over CKM-leading amplitudes. The penguin diagrams are actually
penguin contractions of tree operators [4, 5], and the CKM-leading amplitudes
include also other topologies than tree diagrams, see,  _e.g._ , Ref. [6].

In terms of $U$-spin matrix elements, $r_{\mathrm{QCD}}$ is given as the ratio
of $\Delta U=0$ over $\Delta U=1$ hadronic matrix elements. $U$-spin is a
SU(2) subgroup of the approximate SU(3)-flavor symmetry of the QCD Lagrangian
under unitary rotations of the light quarks $u$, $d$ and $s$. $U$-spin is the
symmetry group which connects $d$ and $s$ quarks. Another such subgroup is
isospin, which connects $u$ and $d$ quarks.

Assuming the SM and $\delta_{\mathrm{QCD}}=\mathcal{O}(1)$ from large
rescattering effects, Eq. (1) implies [7]

$\displaystyle r_{\mathrm{QCD}}^{\mathrm{Exp}}$
$\displaystyle=\mathcal{O}(1)\,.$ (9)

The key question in order to answer the question whether Eq. (1) implies BSM
physics or not is therefore equivalent to the question for the size of
$r_{\mathrm{QCD}}$ in the SM. However, in hadronic charm decays, it is not
clear if methods known from kaon and $b$ physics work. The key issue is if we
can overcome soft QCD in charm decays. At the relevant scale, the strong
coupling $\alpha_{s}$ is too large in order to make a meaningful expansion.
Also, in general an expansion in $\Lambda_{\mathrm{QCD}}/m_{c}$ does not work.
Note that in some cases this might be different, like for the charm hadron
lifetimes, where heavy quark methods show promising results [8, 9]. For
hadronic decays we are however in a different position. For these, we need to
work with a smaller toolbox and find new strategies in order to interpret the
data.

Eq. (9) is consistent with large $\mathcal{O}(1)$ non-perturbative effects
from low energy QCD, which result in [4, 7, 10]

$\displaystyle r_{\mathrm{QCD}}^{\mathrm{SM}}=\mathcal{O}(1)\,,$ (10)

in agreement with expectations based on enhancements that we also see in the
ratio of $\Delta I=1/2$ over $\Delta I=3/2$ $D\rightarrow\pi\pi$ isospin
matrix elements [11], see Ref. [12] for a recent update. On the other hand,
works based on light cone sum rules (LCSR) predict [13, 14, 15]

$\displaystyle r_{\mathrm{QCD}}^{\mathrm{SM}}=\mathcal{O}(0.1)\,.$ (11)

Strikingly, this prediction is one order of magnitude below the experimental
determination in Eq. (9). Another promising ansatz to treat hadronic charm
decays makes use of the available $\pi\pi$/$KK$ rescattering data, see Refs.
[11, 16, 17] for detailed discussions. First ideas on the lattice can be found
in Ref. [18], however the amount of intermediate states proves challenging.

In the following, we review some of the recent developments. In Sec. 2, we
explain the methodology for the extraction of the penguin over tree ratio from
data with theory uncertainties of $\sim 1\%$. In Sec. 3 we discuss how recent
measurements of CP asymmetries of singly-Cabibbo suppressed decays allow to
probe the breaking of the $U$-spin symmetry in CKM-subleading amplitudes. In
Sec. 4 we summarize recent insights on sum rules at higher order in the
$U$-spin expansion. We conclude in Sec. 5.

##  2 Penguin over tree extraction with $\sim 1\%$ theory uncertainty

As mentioned above, the extraction of the penguin over tree ratio Eq. (9) from
$\Delta a_{CP}^{\mathrm{dir}}$ depends on the assumption that the relevant
strong phase is $\mathcal{O}(1)$, see,  _e.g._ , Ref. [7]. For a precision
extraction of $r_{\mathrm{QCD}}$ it would of course be advantageous not to
rely on such an assumption. It is known that the strong phase between the
interfering amplitudes can be obtained in principle from measurements of time-
dependent CP violation or quantum-correlated decays [7, 19, 20, 21, 22, 23,
24, 25]. A difficulty in charm decays is the very slow oscillation between the
mesons. Unlike the time-dependence of $B$ decays, in charm decays we aim just
for the linear term of the time dependence, $\Delta Y^{f}$, in

$\displaystyle A_{CP}(f,t)\approx a_{CP}^{f}+\Delta
Y^{f}\frac{t}{\tau_{D^{0}}}\,.$ (12)

Current determinations of $\Delta Y^{f}$ are compatible with zero [26, 27].
The slopes $\Delta Y^{f}$ of different decay channels with final states $f$
have a universal and a non-universal contribution and the splitting of $\Delta
Y^{f}$ between $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ is formally suppressed at the second power of
$U$-spin breaking effects [28].

The case of $D^{0}\rightarrow\pi^{+}\pi^{-}$ is especially interesting as here
we currently see hints for very large CP violation [26]. In the
$D\rightarrow\pi\pi$ system we have the additional advantage of isospin
symmetry relating the observables of $D^{0}\rightarrow\pi^{+}\pi^{-}$,
$D^{0}\rightarrow\pi^{0}\pi^{0}$ and $D^{0}\rightarrow\pi^{+}\pi^{0}$. Isospin
breaking effects are expected only at the $1\%$ level. Assuming the SM,
isospin allows the determination of the relative strong phase between penguin
and tree amplitudes from direct CP asymmetries and branching ratios only.
Corresponding closed-form expressions have been derived recently in Ref. [12].
$D\rightarrow\pi\pi$ has the same underlying group-theory structure as
$B\rightarrow\pi\pi$ [29], however, the different approximations that apply to
charm and $B$ decays result in different implications for how the underlying
theory parameters relate to direct CP asymmetries and branching ratios. For
example, an important feature of charm decays is that the physics of branching
ratios is to a very good approximation only governed by the CKM-leading
amplitude because the CKM-subleading amplitude is strongly suppressed. This
leads to a decoupling of the physics of branching ratios and CP asymmetries.

An interesting feature of $D\rightarrow\pi\pi$ is that due to the underlying
isospin symmetry, knowledge about the penguin-over-tree ratio in one decay
channel translates into a prediction about this ratio in the other one, namely
[12]

$\displaystyle\frac{r^{00}}{r^{+-}}$
$\displaystyle=\sqrt{\frac{1}{2}\frac{\mathcal{B}(D^{0}\rightarrow\pi^{+}\pi^{-})}{\mathcal{B}(D^{0}\rightarrow\pi^{0}\pi^{0})}\frac{\mathcal{P}^{00}}{\mathcal{P}^{+-}}}\,,$
(13)

where $\mathcal{P}^{f}$ are the corresponding phase space factors and $r^{00}$
and $r^{+-}$ are the penguin over tree ratios of the isospin decomposition of
$D^{0}\rightarrow\pi^{0}\pi^{0}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays,
respectively. With current data it is found [12]

$\displaystyle r^{+-}$ $\displaystyle=5.5^{+14.2}_{-2.7}\,,$ (14)
$\displaystyle r^{00}$ $\displaystyle=5.2^{+13.3}_{-2.4}\,,$ (15)

_i.e._ , the central values are unexpectedly large. With future data from LHCb
Upgrade II and Belle II it will be possible to reduce the uncertainties
considerably [12]. Once also time-dependent CP violation measurements are
available, the above methodology can be used in order to not only determine,
but overconstrain the $D\rightarrow\pi\pi$ system and thereby probe for BSM
physics.

##  3 Testing the $U$-spin symmetry in CKM-subleading amplitudes

Recent new measurements by LHCb went beyond just the measurement of the
combination of CP asymmetries in Eq. (1) and provide separate determinations
of the involved CP asymmetries [26]

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow K^{+}K^{-})$
$\displaystyle=(7.7\pm 5.7)\cdot 10^{-4}\,,$ (16) $\displaystyle
a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow\pi^{+}\pi^{-})$ $\displaystyle=(23.2\pm
6.1)\cdot 10^{-4}\,,$ (17)

implying the first evidence for CP violation in a single decay channel. The
central values of Eqs. (16) and (17) violate the $U$-spin limit sum rule [30]

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow
K^{+}K^{-})+a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow\pi^{+}\pi^{-})$
$\displaystyle=0\,.$ (18)

While a violation of Eq. (18) is expected, the amount goes beyond the generic
expectation of $\sim 30\%$ SU(3)F breaking, however yet with large errors.
With future data, additional sum rules such as [30]

$\displaystyle a_{CP}^{\mathrm{dir}}(D_{s}^{+}\rightarrow
K^{0}\pi^{+})+a_{CP}^{\mathrm{dir}}(D^{+}\rightarrow\overline{K}^{0}K^{+})=0$
(19)

can be tested in order to study if a consistent picture emerges [25]. Improved
versions of the sum rules Eqs. (18) and (19) connect the decays

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow K^{+}K^{-})\,,\quad
a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow\pi^{+}\pi^{-})\,,\quad
a_{CP}^{\mathrm{dir}}(D^{0}\rightarrow\pi^{0}\pi^{0})\,,\quad\text{and}$
$\displaystyle a_{CP}^{\mathrm{dir}}(D^{+}\rightarrow K_{S}K^{+})\,,\quad
a_{CP}^{\mathrm{dir}}(D_{s}^{+}\rightarrow K_{S}\pi^{+})\,,\quad
a_{CP}^{\mathrm{dir}}(D_{s}^{+}\rightarrow K^{+}\pi^{0})\,,$

respectively, and can be found in Ref. [31]. The latter highlight the synergy
and complementarity of LHCb [1, 32, 26] and Belle/Belle II [33, 34].
Explanations of large $U$-spin breaking in the CKM-subleading amplitudes also
include $Z^{\prime}$ models [35, 36], as these generate operators with the
flavor content $\bar{s}c\bar{u}s$ and $\bar{d}c\bar{u}d$ with non-universal
coefficients. Ref. [36] presents viable models with a leptophobic $Z^{\prime}$
below $O(20\,\text{GeV})$, including a pattern of CP violation in
$D\rightarrow\pi\pi$ that includes a violation of the isospin sum rule [23,
37]

$\displaystyle a_{CP}^{\mathrm{dir}}(D^{+}\rightarrow\pi^{+}\pi^{0})$
$\displaystyle=0\,.$ (20)

More study is needed of the puzzling violation of Eq. (18) once more data
becomes available.

##  4 Solving the Problem of Higher Order $U$-Spin

The discussion in Sec. 3 motivates to go to higher order in the $U$-spin
expansion in order to derive sum rules that hold to higher precision than Eq.
(18). Going to higher order in SU(3)F breaking is complicated by the
proliferation of parameters, visibly already at next to leading order [38]. In
Ref. [39] we derived new theorems which enable calculations to arbitrary order
in the $U$-spin expansion on the amplitude level. This allows especially also
to determine _a priori_ up to which order sum rules actually exist. The new
method uses symmetry properties of Clebsch-Gordan coefficients and the fact
that any given system can be reduced to the discussion of a doublet-only
system out of which the given system is constructed.

Many more opportunities lie ahead in this direction. We expect that this
methodology will be especially useful for multi-body decays. Precision sum
rules at high order are likely for these to exist due to the higher
combinatorial possibilities in comparison to two-body decays.

The first step regarding three-body decays however will be to probe the ratio
of $\Delta U=0$ over $\Delta U=1$ matrix elements of underlying pseudo two-
body $D\rightarrow VP$ decays in order to see if their order of magnitude
agrees with the analogous ratio for $D\rightarrow PP$ decays in Eq. (9) [40].
This enables an important consistency check of the hierarchies of $U$-spin
matrix elements. Such endeavors are complementary to model-independent
searches for CP violation in multi-body decays for example with the energy
test [41, 42, 43]. Another frontier are charmed baryon decays, see recently
Refs. [44, 45, 46, 47], to which the methodology of Ref. [39] also applies.

##  5 Conclusions

This is just the beginning of the exploration of charm CP violation. Until
recently we had only a measurement of $\Delta a_{CP}^{\mathrm{dir}}$. Now,
with the evidence of CP violation in $D^{0}\rightarrow\pi^{+}\pi^{-}$ we have
two data points. In order to benefit from symmetry-based methods for the
interpretation of the data, we need more measurements of CP asymmetries of
singly-Cabibbo suppressed decays. Only then we will be able to test the
corresponding sum rules. Time-dependent measurements will be very important to
use isospin methods to overconstrain the $D\rightarrow\pi\pi$ system. The
development of new higher order sum rules will provide opportunities to
benefit from the abundant data on multi-body decays. The key question if we
can tell a loop from a tree remains a challenge for charm theory, but no
matter what, we will learn something new: about QCD, and maybe also about new
physics.

## Acknowledgments

S.S. is supported by a Stephen Hawking Fellowship from UKRI under reference
EP/T01623X/1 and the STFC research grants ST/T001038/1 and ST/X00077X/1.

## References

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