# Acoustic surface modes on metasurfaces with embedded
next-nearest neighbor coupling

D. B. Moore<EMAIL_ADDRESS>J. R. Sambles  A. P. Hibbins  T. A. Starkey
G. J. Chaplain  Centre for Metamaterial Research and Innovation, Department of
Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom

###### Abstract

We design, simulate, and experimentally characterize an acoustic metasurface
comprising of a 1D array of open, sound-hard, cavities, modulated with beyond-
nearest-neighbor (BNN) couplings in the form of additional connecting cavities
embedded beneath the surface. The hidden complex structure is realized readily
with additive manufacturing techniques (3D printing). The dispersive
properties of the supported localized acoustic surface waves are influenced by
competing power-flow channels provided by the BNN couplings, that generate
extrema in the dispersion spectra within the first Brillouin zone. The
structure supports negatively dispersing ‘backwards’ waves that we
experimentally verify. Such structures thereby provide a route to enhanced
acoustic sensing by acoustic metasurfaces.

##  I I Introduction

Dispersion engineering is integral to the design of acoustic metasurfaces,
with many applications across sensing, acoustic antennas, acoustic lensing,
and tailored sound absorption [1, 2, 3, 4, 5, 6, 7], in addition to passive
beam steering with topological surfaces [8, 9, 10, 11]. The wider theme of
engineering dispersive materials has been a primary focus of metamaterial
research for over a decade [12, 13, 14, 15, 16]. Typically, acoustic
metasurfaces are structured planar geometries, tessellated in some periodic
distribution that manipulates acoustic waves through the collective
interaction of resonators.

Figure 1: (a-b) FEM calculated dispersion relations (frequency against in-
plane wave vector, $k_{||}$) showing the acoustic surface waves supported on a
sample comprised of open hole cavities utilising NN coupling (a), and a
revised surface containing BNN coupling channels within the substrate (b).
Both samples have a periodicity $\lambda_{p}$ = 10 mm along the $x$-axis and
are patterned with cylindrical resonators with depth $d$ = 30 mm and radius
$r$ = 2 mm. The component of acoustic intensity ($I_{||}$) in the direction of
propagation for each eigenmode is indicated by the color-map. The solid black
line represents the sound-line ($\boldsymbol{k}_{0}$). Insets show example
geometries with periodicity $\lambda_{p}$ in the propagation direction ($x$),
depth $d$, and diameter $2r$.

The canonical example of a simple acoustic metasurface is a series of closely,
periodically spaced resonant cavities; the resonators diffractively couple to
their nearest neighbors (NN) through near-field effects, allowing an acoustic
surface wave (ASW) to form, localized at the interface between the metasurface
and the surrounding fluid. The formed ASW will propagate along the array of
cavities, decaying evanescently away from the surface due to the effective
impedance condition of the rigid cavities patterning the surface [17]. The
substrate is assumed to be sound-hard, contributing no elastic motion. This
example structure, with a single resonator per unit cell, supports only a
single ASW associated with the cavity’s fundamental resonance (shown in Fig.
1(a)), and forms an acoustic analogue to electromagnetic spoof-surface
plasmons on perfectly conducting patterned substrates [18]. Increasing the
number and complexity of the modes supported on the surface requires amending
the structure factor, that is, the geometry and relative dimensions of the
comprising resonant elements; the number of modes is linked to the degrees of
freedom within the unit cell [19, 20, 21]. An approach to engineering
dispersion is to employ more complex structure factors that incorporate
beyond-nearest-neighbor (BNN) coupling, via the introduction of additional
resonators that acoustically connect a unit cell to its $L^{\text{th}}$
nearest-neighbor ($L>1$), without altering the periodicity of the structure
[22, 23, 24, 25]. In a 1D NN-only coupled system, defined by a reciprocal
lattice vector of size $2\pi/\lambda_{p}$, with $\lambda_{p}$ the periodicity,
the typical dispersion is defined by modes that approach the 1st Brillouin
Zone Boundary (BZB) (where the in-plane wave vector,
$k_{\parallel}=\pi/\lambda_{p}$) with corresponding vanishing group velocity,
$v_{g_{||}}=\frac{\partial\omega}{\partial k_{||}}$; a standing wave is formed
along the surface, satisfying the Bragg condition [26]. In NN-only coupled
systems there is therefore conventionally a single extremum in the dispersion
curves when the mode reaches the asymptotic limit at the BZB. More complex
structures that include symmetry broken geometries (either spatially, or in
time) are some exceptions to this generalization [27]. For systems with BNN
coupling, it has been known for some 70 years that there will be $L-1$ extrema
within the 1st Brillouin zone, causing regions in the dispersion curves to
have alternating signs of the group velocity [28, 24], giving rise to modes
that are said to have negative dispersion and are often termed ‘backwards
waves’; the direction of wave propagation (momentum) and energy-flow are anti-
parallel [29].

Phenomena involving negative dispersion has been realized for some time with
electromagnetic metasurfaces [30, 31, 32, 33], and recently BNN (sometimes
termed ‘non-local’ [34]) coupling in acoustic and elastic metamaterials has
received renewed attention [22, 35, 25, 24]. Martinez et al. [35] showed
experimentally in both elastic and acoustic regimes that in a 3D structure
comprised of BNN coupling to the $L=3$ neighboring cell, a single fundamental
mode is supported, exhibiting a maximum and minimum in its dispersion before
the $1^{st}$ BZB. Here we extend these concepts to the canonical cavity
acoustic metasurface by incorporating a hidden (i.e. embedded beneath the
metasurface) next-nearest neighbor ($L=2$) coupling in the form of channels
that reach-around the central open cavities. Nearest-neighbor coupling exists
through near-field diffraction along the top and bottom of the surface, in
addition to beyond-nearest neighbor coupling provided by the concealed
connectors (shown in Fig. 1(b)). The additional structure provides an
alternate route for power flow, and it is the competition between these two
coupling paths that causes the extrema in the dispersion within the 1st BZ.
The result is regions in $k_{\parallel}$-space where there is a reversal in
net power flow along the direction of propagation i.e. the surface supports
negatively dispersing waves.

Given the spatial symmetries associated with the unit cell, an anti-symmetric
mode is supported in addition to the fundamental mode, see Fig. 1(c). The
number of maxima can be tuned arbitrarily as a function of linking
${L}^{\text{th}}$ neighboring unit cells as well as breaks in the structures
symmetry.

We demonstrate these effects experimentally for ASWs, using rapid prototyping
(3D printing) to produce a scalable sample. This is an ever-increasingly
popular method for realising complex, small-scale metamaterial geometries that
would otherwise be difficult and expensive to manufacture [36]. In what
follows we demonstrate that the observed negative dispersion is genuine in
that it results from the competition of power flow rather than diffractive
effects associated with, for example, glide symmetry [37, 20].

The structure of this paper is as follows. Firstly, in Section IIII, the
metasurface geometry demonstrating BBN coupling is described and the supported
eigenmodes evaluated, using Finite Element Method (FEM) numerical methods
[38]. We introduce a second geometry, with a perturbed dispersion relation, by
altering the boundary conditions along the metasurface and thereby breaking
geometric symmetries associated with the structure. 3D-printing fabrication
methods are then detailed, before the experimental results are presented in
Section IIIIII.

##  II II Beyond Nearest Neighbors Coupling for Acoustic Surface Waves

Figure 2: (a,b) Original and covered design unit cells. The sample has a
periodicity $\lambda_{p}$ = 10 mm along the $x$ axis, and is patterned with
cylindrical resonators with depth $d$ = 30 mm and radius $r$ = 2 mm. The
internal coupling channels also have radius $r$, separated span-wise across
the unit cell by width $w$ = 12.5 mm and vertically from the centre by height
$h$ = 9.75 mm. b Illustrates the sound-hard boundary positioned along the
covered side of the sample. (c,d) FEM calculated dispersion relations of
eigenmode solutions showing the acoustic surface waves within the original (c)
and covered (d) designs. The acoustic intensity ($I_{||}$, normalized to the
maximum of the lowest band) in the direction of wave propagation for each
eigenmode is indicated by the color-map. Instantaneous (normalized) pressure
fields $P$ (with $P_{b}=\left|{P-P_{0}}\right|$, where $P_{0}$ is the
background pressure field) of eigenmodes taken at the 1st BZB
($k_{\parallel}=X\equiv\pi/\lambda_{p}$) are shown for the original unit cell
design (a).

###  II.1 Design

The design process from a nearest-neighbor-only acoustic metasurface, to one
with additional BNN coupling. Figure 1(a) shows the simplest unit cell
comprising an acoustic metasurface, a single periodic row of holes (cavities),
of radius $r=2$ mm, and depth $d=30$ mm, periodically spaced by
$\lambda_{p}=10$ mm, coupled through NN near-field effects (diffractive
coupling). The lowest frequency ASW supported by the structure is associated
with the fundamental resonance of the structure (cavities) with limit
frequency given approximately by $f_{res}=c/4(d+\Delta L)$, where $c$
represents the adiabatic speed of sound in air and $\Delta L$ represents the
end correction for a pair of openings ($\Delta L=\frac{8r}{3\pi}\approx 0.8r$)
[39, 40]. The in-plane wave vector of the surface mode, $k_{\parallel}$, is
indistinguishable from the free-space wave or sound-line with $k_{0}=\omega/c$
for small values of $k_{\parallel}$, for wave of radian frequency $\omega$. As
$k_{\parallel}$ increases, the mode progressively disperses away from the
sound-line, becoming strongly bound to the surface. The mode will disperse,
with collinear phase and group velocity, towards its asymptotic limit until it
meets the 1st BZB with zero group velocity where it forms a standing wave see
Fig. 1(a).

To explore BNN coupling, the complexity of the structure factor needs to be
increased. Increasing the number of resonators within the unit cell (thereby
increasing the structure factor) allows additional modes to be supported by
the surface [41] i.e. we are considering the complexity of the modes in terms
of only the lowest order resonances, and not the infinite series supported by
a single cavity. In this context, the complexity of such a structure is
relatively low; all of the additional resonators are confined to the unit
cell. Alternatively, cylindrical resonators (i.e. tubes within the structure)
that link the $\pm~{}L^{th}$ nearest-neighbors, whilst bypassing the resonant
elements comprising the cells between can be introduced, and hence provide a
route to BNN coupling. The structure factor can be further modified through a
rotational translation of the BNN resonators about the centre of the open
cavity, producing the cross hatch structure shown in Fig. 2(a,b) and 3. The
unit cell, at its lowest definable order of symmetry, now contains a resonator
that passes through the cell, thereby only interacting acoustically with the
other constituent resonators in the next-nearest cell [42, 43, 28], see Fig.
2(a). The additional coupling channels have a fundamental resonance dictated
by their length (typically lower in frequency than the open cavity due to
their increased length). The BNN coupling manifests due to the overlap of the
evanescent fields from these additional connectors with those of the open
cavity.

The dispersion of the supported modes drastically changes from the case of a
single resonant cavity, with extrema in the lowest order mode now existing
within the first BZ, shown in Fig. 2; a feature commonly associated with
symmetry breaking or topological effects [27, 11]. This implies that over
certain regions of $k_{\parallel}$, the group velocity of the mode is
negative, and the net power flow is travelling opposite to the direction of
wave propagation (noted by the regions of negative intensity in Fig. 1(b));
here, the integral of the intensity ($I_{||}$) is calculated along the
direction of periodicity ($x$-axis) throughout the whole computational domain,
providing the direction of the net flow of energy of the ASW. In the frequency
domain the intensity is given by

$\boldsymbol{I}=\frac{1}{4}\left(p\boldsymbol{v}^{*}+p^{*}\boldsymbol{v}\right),$
(1)

where the velocity, $\boldsymbol{v}$ is related to the pressure $p$ through
$\boldsymbol{v}=\frac{-1}{i\omega\rho}\nabla p$, with $*$ denoting complex
conjugation.

Given the spatial symmetries present within the unit cell (where the
connections between the coupling channels and open hole are mirrored about the
centre of the unit cell), an additional anti-symmetric mode is also supported
by the structure due to the coupling of adjacent resonators within the open
cavity, see Fig. 2(c). The anti-symmetric mode is odd about the $xy$ plane
passing through the centre of the structure and disperses with the opposite
sign in $v_{g}$ and $I_{||}$ to the fundamental mode, with the modes
exchanging sign of the net power flow when
$k_{\parallel}=\frac{\pi}{L\lambda_{p}}$, see Fig. 2(c). The fundamental mode
approaches the BZB beneath the anti-symmetric mode, as both the coupling
channels within the unit cell are resonating in-phase relative to one another,
thus having a lower frequency, see Fig. 2(c). For the anti-symmetric mode, at
the band edge, both channels are resonating in anti-phase ($\pi$ radians
difference) relative to one another, thereby halving the length of the
resonator compared to that of the fundamental, hence increasing the resonant
frequency, see Fig. 2(c).

The propagation direction of the wave vector (either positive or negative) can
be inferred from probing the near-field, since the evanescent fields of the
ASW decay into the surrounding medium via the open cavities. The number of
extrema in the dispersion before the 1st BZB can be tuned by altering $L$
[28]. In this study, the value of $L$ has been set to 2 (or next-nearest
neighbor). Provided the required number of coupling channels can be contained
within the unit cell while remaining acoustically isolated, any integer value
for $L$ can be conceived.

Figure 3: Rendered schematics of the sample, areas colored gold represent PLA,
and grey represents water soluble PVA. The sample is comprised of two halves,
each of 20 unit cells.

The symmetry of the unit cell can be broken by replacing one of the open ends
of the open cavity with a sound-hard boundary condition, causing the
fundamental and anti-symmetric modes to anti-cross, forming a bandgap, see
Fig. 2(d). By covering one of the surfaces the coupling strength of the BNN
channels closest to the closed end increases, biasing the power flow through
that junction. The same number of extrema are present within the 1st BZ.
Additionally, similar results are found when reversing the direction of the
BNN connections on one side of the unit cell, analogous to a mirror plane in
the centre of the unit cell.

###  II.2 Fabrication: 3D printing

The sample was 3D printed using an Ultimaker S5, extruding polylatic acid
(PLA) filament. Support structures (for the inner tubes) were used to aid the
printing process, created using a water soluble filament, polyvinyl alcohol
(PVA) which was removed before measurements. The sample has length
$L_{\text{total}}=400$ mm, periodicity $\lambda_{p}$ = 10 mm along the $x$
axis (40 unit cells total), and is visibly patterned with cylindrical
resonators with depth $d=30$ mm and radius $r=2$ mm, the coupling channels
have the same radius, see Fig. 2(a,b) for schematic and Fig. 3 for render. The
sample was manufactured with a 40% infill density to maintain a sound-hard
boundary condition at the walls of the sample. The scale of the sample
dimensions was carefully chosen to produce resonances optimally situated for
detection experimentally; the frequency range of the sample can be tuned by
scaling thegeometry dimensions.

##  III III Experimental Results

To verify the numerical predictions we experimentally characterized the ASW
fields supported on the surface of the sample. Model dispersion relations as a
function of the in-plane wavenumber, $k_{||}$, were obtained with COMSOL
Multiphysics (version 5.6) [38] using the Finite Element Method. Dispersion
relations shown are eigenmodes of a unit cell (see Fig. 2(a,b)) with Floquet-
periodic boundaries to represent an infinite sample. This model assumes the
resonators are cylindrical and perforate/enclosed by an acoustically rigid
surface. The dispersion for each excitation arrangement was characterized by
measuring the acoustic near-field using a probe microphone mounted on a
motorized xy scanning stage. Samples were excited by a pair of Visaton SC5 13
mm tweeters mounted within conical attachments with 3 mm exit diameters. The
exits of the conical housings were attached to cylindrical waveguides of
length 150 mm, positioned within the coupling channels to allow the sources to
be correctly positioned due to their size, see Fig. 4. The pair of
loudspeakers could be driven in- or out-of-phase relative to one another. To
minimize signal from direct radiation, the needle microphone (Brüel & Kjær
Probe Microphone type 4182) was positioned 0.5 mm over the centre of the open
cavities. The surface was scanned at a resolution of 0.45 mm for a scan length
of 400 mm ($x$-axis). A single cycle 5 kHz Gaussian envelope pulse (broadband)
was used to excite the sample at each microphone position. An average was
taken over eight measurements at each spatial position to improve the signal-
to-noise ratio. The dispersion was characterized by performing a Fast Fourier
Transform (FFT) on the measurement data to obtain the frequency (Fourier)
amplitude of the real spatial component as a function of wave vector,
$|k_{||}|$, thereby recovering the single-sided spectra whilst we only measure
power flow away from the source towards the detector.

Figure 4: Render illustrating the acoustic measurement experiment, a pair of
loudspeakers mounted within conical housings and attached to cylindrical
waveguides are shown, positioned at the end of the sample and the microphone
positioned over the open cavities.

Figure 5 shows the measured dispersion of the modes of the sample, with and
without altered symmetry, through the logarithm of the experimental Fourier
amplitude (real FFT). Results from a loss-inclusive FEM model are overlaid as
blue points, for both excitation field arrangements and each symmetry
arrangement; with the left hand side of the dispersion relations from in-phase
excitation of the sources and the right hand side showing the results from the
out-of-phase excitation. Figure 2(c) shows the pressure field solutions for
the anti-symmetric mode to have a zero pressure state between the opposing
coupling channels ($\pi$ phase difference). The pressure must reach zero
between the opposing resonators to satisfy the phase arrangement of the
eigenmode. However, despite the zero pressure condition, the mode is
detectable due to the pressure amplitude of the decaying fields above the
surface. This is evidenced by the detected anti-symmetric modes in Fig. 5.
Analysing the pressure fields within the structure at the first extremum in
the dispersion, the increased length of the coupling structures becomes
apparent, as the pressure oscillations are confined to the coupling channels,
while the free-space wavelength is still four times the lattice pitch. The
resonance of the open cavities is not excited until the $\lambda/2$ condition
of the substrate depth is satisfied. The fundamental mode leaves the sound-
line with a near-constant group velocity; this frequency at which the mode
leaves the sound-line, transitioning from the free-space speed of sound to
that of a waveguide mode [37]. The reduction of the sound-speed in this region
is reminiscent of coiled space metamaterials that provide a longer acoustic
path length [44]. Exciting the sample with a pair of monopole sources allows
the phase and amplitude of the excitation field to match that of the mode. In
Fig. 5(b) the amplitude of the fundamental mode is decreased when the sample
is excited with an excitation field which matches the anti-symmetric mode
shape.

Figure 5(b) shows the measured dispersion of the sample with altered symmetry.
Covering one side of the surface prevents energy from decaying into the
surrounding air, increasing the amplitude through that junction, biasing the
power flow. Replacing one of the open boundary conditions with a sound-hard
condition alters the symmetry of the modes, lifting the degeneracy, forming a
low-frequency bandgap across the 1st BZ. The resonant condition of the cavity
becomes approximately $\lambda/4$ due to the sound-hard boundary condition at
the bottom of the open hole, decreasing the frequency of the fundamental
resonance.

Figure 5: Comparisons between FEM eigensolutions (cyan points) and logarithmic
experimental Fourier spectra (color scale), obtained via the real FFT. Left
column shows results for in-phase excitation (red points in schematic above
the dispersion curves representing the source positions; right column for the
out-of-phase excitation (red and blue points in schematic), for (a) original
set-up (b) with one side covered. Note we plot $|k_{||}|$ on the $x$-axis.

##  IV IV Conclusions

We have manufactured a sample using 3D printing which is visibly patterned
with a 1D array of resonant cavities while embedding within the substrate, a
network of coupling channels facilitating next-nearest-neighbor coupling. The
eigenmodes supported by this surface have been calculated using numerical
modeling and demonstrated experimentally. We have detected, through probing
the near-field, the full dispersion spectra across the frequency range
considered by matching the excitation symmetry to that of the supported
eigenmodes. The dispersion of the supported modes is tuneable through
manipulating the symmetry of the unit cell by covering select resonant
cavities. Further frequency bands beyond those explored in this study can be
achieved through scaling the geometry.

## Data availability

All data created during this research are openly available from the University
of Exeter institutional repository at https://ore.exeter.ac.uk/.

## acknowledgements

The authors wish to acknowledge financial support from the Engineering and
Physical Sciences Research Council (EPSRC) of the United Kingdom, via the
EPSRC Centre for Doctoral Training in Metamaterials (Grant No. EP/ L015331/1).
D.B.M., T.A.S., A.P.H. and J.R.S. acknowledge the financial support of DSTL.
G.J.C. gratefully acknowledges financial support from the Royal Commission for
the Exhibition of 1851 in the form of a Research Fellowship.

## Competing interests

The authors declare no competing interests.

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# Raply: A profanity-mitigated rap generator

Omar Manil BENDALI, Samir FERROUM, Ekaterina KOZACHENKO, Youssef PARVIZ,
Hanna SHCHARBAKOVA, Anna TOKAREVA, Shemair WILLIAMS

###### Abstract

The task of writing rap is challenging and involves producing complex rhyming
schemes, yet meaningful lyrics. In this work, we propose Raply, a fine-tuned
GPT-2 model capable of producing meaningful rhyming text in the style of rap.
In addition to its rhyming capabilities, the model is able to generate less
offensive content. It was achieved through the fine-tuning the model on a new
dataset $Mitislurs$, a profanity-mitigated corpus. We evaluate the output of
the model on two criteria: 1) rhyming based on the rhyme density metric; 2)
profanity content, using the list of profanities for the English language. To
our knowledge, this is the first attempt at profanity mitigation for rap
lyrics generation.

##  1 Introduction

Since its inception in the late 1980’s rap music has become one of the most
influential music genres owing to its ability to address various ethical and
social issues affecting people worldwide.

In this paper, we study the problem of determining the most lyrically relevant
rap line while taking into account ethical issues. Our interest in this
problem is motivated by the following perspectives. Firstly, we want to study
the structure of rap lyrics to generate relevant text which adheres to the
features of rap lyrics. Secondly, we are aware of social and ethical issues
around this topic, but we also should treat respectfully the specific range of
language expressions that is often characteristic of the rap genre. Thirdly,
we want to be able to give rappers the ability to create expressive works
about topics relevant to their experience without the need to fall back on
human ghostwriters when the works they create are not lyrically strong (i.e.
lack of rap techniques).

We formalize this problem as an auto-completion task. Given a query
$p=[A:,s(1),s(2),...,s(n)]$, an auto completion returns
$q=[B:,t(1),t(2),...,t(n)]$, where q is an extension of p. In this specific
rap lyrics generation problem, $p$ is the set of the input tokens $s(i)$, and
$q$ is the set of model’s generated tokens $t(n)$ that shares syntactic,
semantic and rap-specific characteristics with $p$ (rhyming), we assume that
$A$ marks the beginning of the user input line and $B$, the beginning of the
model’s generated line.

Our main contributions can be summarized as follows:

  1. 1.

We propose a method for mitigating the number of profanities in rap lyrics
generation through the construction of the $Mitislurs$ corpus, in order to be
mindful of the issues often discussed in the rap genre.

  2. 2.

We introduce a method of rap lyrics generation that is able to produce
satisfying rap in terms of rhyming.

Content Warning. This paper contains material that may be offensive or
upsetting. In contrast to the Appendix A, where the abusive language examples
are provided verbatim, examples in the main text are represented by initial
letters of the profanities followed by asterisks. The authors oppose the use
of abusive language.

##  2 Related work

Automatic rap lyrics generation is a challenging task in natural language
processing. It requires a system to be able to learn complex linguistic and
semantic information from lyrics and to use this knowledge to create a new,
creative song.

This task has become a more active research area in recent years, mainly due
to the success of deep learning techniques (Potash et al., 2015; Malmi et al.,
2016; Manjavacas et al., 2019; Nikolov et al., 2020; Liu et al., 2022).

Potash et al. (2015) show the effectiveness of an LSTM model for rap lyrics
generation: using its own rhyme scheme, line and verse length, Ghostwriter can
generate lyrics similar to the rap style of a specific rapper. In Malmi et al.
(2016) the authors address the problem of rap lyrics generation as an
information retrieval task: the system treats input line as a query to which
it predicts the most suitable next line. The Transformer-based Vaswani et al.
(2017) approach to the generation of rap using content words from the song is
described in Nikolov et al. (2020). Liu et al. (2022) proposed a ChipSong – a
lyrics generation system that enables the user to control different
attributions of the generated song such as word-level format control and rhyme
control.

We also considered studies related to poem generation, as they share similar
objectives in terms of rhyme production. A study by Lo et al. (2022)
introduced GPoeT-2 model, based on GPT-2 architecture and capable of
generating from any given subject. Lastly, Hämäläinen et al. (2022), also
sought to establish a model capable of generating French poems by using two
pre-trained language models.

However, to the best of our knowledge, none of the previously mentioned
studies take into account the presence of profanity, although based on a study
conducted by Frisby and Behm-Morawitz (2019), among the many musical genres,
rap is the one that uses the most profanities compared to other genres.
Considering that large language models demonstrate bias and toxicity (Gehman
et al., 2020; Sheng et al., 2019; Wallace et al., 2019), it can be assumed
that the resulting output from the models may contain offensive language.

##  3 Rhyming and rap

Rhyme may be defined as the “acoustic agreement of sounds (vowels and
consonants), words or groups of words” Grofčíková and Máčajová (2020). In
other words, rhyme is simply having a similar sound between two or more words.
Rhyming is a defining feature of various works of art such as poetry,
songwriting as well as rap.

###  3.1 Types of rhymes

Rhymes may take various forms, such as perfect rhymes or slant rhymes, to name
a few. Perfect rhymes are one of the most popular and well-known types of
rhyming which involves words sharing the same syllabic sound stress occurring
at the end of the word. The sound similarity and syllabic stress between the
words walk and talk, stalk are examples of perfect rhymes. While slant rhymes,
which is a differentiation of perfect rhymes, do not have such a strict sound
pattern, and the rhyme is often located internally within the words. Take for
example the words, home and none, the similarity exists internally within each
word as the word stress happens at the beginning of both words resulting in a
rhyme.

Other not-so-common forms of rhyme include assonance which involves the
repetition of vowels for example, drown and sound. Consonance, another form of
rhyme is characterized by rhyming through the repetition of consonant sounds
such as staff and flat. Many of these styles of creating rhyme overlap or work
together to create rhythm. For instance, alliteration is a combination of
assonance and consonance where the same consonant or vowel is repeatedly
stressed in each word. The sentence Peter Piper picked a peck of pickled
pepper, is a perfect example of alliteration at play.

Rhymes may be further defined by their positions within a line or sentence or
verse. End rhymes for instance appear at the end of a sentence or line while
internal rhymes take place within a line or sentence.

###  3.2 Rap song structure

Various artists constantly make use of rhyming schemes to create rhythm and
rhyming patterns, especially in rap music. Many rappers follow some form of
the rhyming scheme in their music and this is quite apparent in the various
rap songs chosen for our corpus. And in like manner, these rhyme patterns may
be found either at the beginning or the end of a line, sentence, or verse.

Rap music makes use of bars, which is simply another name for the lyrics of
the lines the rappers use in their music. These bars are then grouped by what
are known as verses. Verses refer to the different segmentations that occur in
a rap song. In general, rap music employs about 16 bars per verse. Each verse
used by the rapper tells some form of a story and is dependent on some form of
rhyming scheme. In addition to this, many verses are preceded by what is known
as a hook or the punchline of the verse. This is the part of the verse which
captures and maintains the listener’s attention throughout the entirety of the
lyrics.

##  4 Data collection

In this section, we describe the steps we made to obtain the new corpus and
preprocess it.

To build a system that generates a rap line based on the text input, we
compiled a large corpus of rap lyrics together with lyrics metadata. For this
purpose, we scrapped Genius.com 111https://genius.com/., a large collection of
lyrics, to obtain the necessary data.

Foremost, we assembled a list of 89 American English-speaking rap performers
chosen based on their renowned writing and rhyming skills. Further, via the
free Genius Song Lyrics API request, which is made using a client Access Token
generated on the site, we got access to all available information about that
particular performer. With the help of lyricsgenius library
222https://lyricsgenius.readthedocs.io/en/master/., we collected the required
information, specifically the artist’s name, the name of the song, the year of
release of the song, and the lyrics.

Thus we obtained a raw corpus of 21,801 songs with a size of 14,296,438
tokens.

We use preprocessing which was designed with attention to the specifics of the
data. Based on the structure of the rap song, we removed parts of the songs
that we found unnecessary (e.g., instrumental) and repetitive (e.g., song
remixes). In addition, we removed all the songs written in languages other
than English. In songs, the parts that represent the chorus of songs are
repetitive, and would not help to generate sophisticated lines; also,
interludes, do contain unnecessary discourse that would not be relevant to our
task. There are also confusing parts, that were dropped, which do not exist in
all the songs (e.g., [Intro] or non-ASCII characters). Additionally, we have
chosen to keep only verses parts. The aim of using verses only is to focus
only on parts of the lyrics that contain the most rhyming pieces, other parts
are considered repetitive and irrelevant to the rhymes.

##  5 Data annotation

Modern language models are trained on large text corpora from the Internet. If
a model receives training data containing slurs or abusive language, it will
learn to predict these words during the training step and subsequently produce
the output containing these words. To minimize the occurrence of profanities
in the data generated by the system, we decided to automatically filter data
by offensive language beforehand. To compile the dataset, we used The
Obscenity List, developed by Surge AI 333https://www.surgehq.ai/., which
contains over 1600 popular English profanities and their variations
444https://github.com/surge-ai/profanity., including non-standard spellings
(e.g., canonical form damn could be written as g0ddamn). Figure 1 illustrates
the distribution of profanities by category according to their severity
ratings in The Obscenity List 555Severity ratings distribution [1-3]: from 1
to 1.4 - mild, strong, and from 1.5 to 2.4 - strong, and from 2.5 to 3 -
severe.. It can be seen that the prevailing majority of profanities are sexual
anatomy/ sexual acts, having a strong degree of severity. The second most
frequent group is racial/ethnic profanities, the level of severity of which
is, for the most part, the maximum.

Figure 1: The distribution of profanities by categories according to their
severity ratings in The Obscenity List.

In order to tokenize and lemmatize each lyric from the raw dataset, we used
NLTK library (word_tokenize() to tokenize the texts and the Wordnet Lemmatizer
to lemmatize them). Then we automatically extracted the profanities from each
song. The obtained dataset with offensive language (see Figure 2) includes
lyrics in which 50,130 profanities were found, as well as lyrics with no
profanities at all, that is 2,225 songs.

Moreover, we additionally used two types of annotation, which were produced by
Surge AI 666In the Appendix A are shown some examples of the profanities from
each group.: profanity categories and severity rating.

Figure 2: Distribution of profanities across matched categories in the raw
dataset.

Profanity categories. Firstly, we performed the annotation of each profanity
for one of 11 profanity’s primary categories:

  * •

sexual anatomy / sexual acts

  * •

bodily fluids / excrement

  * •

sexual orientation / gender

  * •

racial / ethnic slurs

  * •

mental disability

  * •

physical disability

  * •

physical attributes

  * •

animal references

  * •

religious offense

  * •

political

  * •

other / general insult

As can be seen in Figure 2, only 8 out of 11 profanity categories from the The
Obscenity List were found in a compiled dataset. The most frequent profanity
category is sexual anatomy / sexual acts.

Severity rating. In The Obscenity List, the severity rating annotation was
done manually by 5 Surge AI experts. The data annotators were asked to rate
how severe they believed each profanity to be, on a 1-3 point scale. The
rating reflected in the dataset is the mean of those 5 ratings. We took this
type of annotation in order to get the values of the $slur\\_score$ for each
lyric, which was calculated using the following formulas:

$ws\\_score(line)=\frac{\Sigma
sev\\_ratings(line)}{number\\_of\\_tokens(line)}$

$slur\\_score=\frac{\Sigma ws\\_score}{number\\_of\\_lines}$

We first compute the weighted severity score ($ws\\_score$) defined as the sum
of the severity rating of each profanity identified in one line and average
over its number of tokens. Then, we average the sum of weighted severity
scores of all the lines of the song over the total number of lines to obtain
the $slur\\_score$. Next, we drop all the songs lyrics that have a
$slur\\_score$ above the threshold value of 0.05 ($3^{rd}$ quantile of the
$slur\\_score$ distribution, as shown in Figure 3). We deliberately choose to
filter slurs on a song level only, because of the possibility of breaking
lyrical structure and cohesion.

Figure 3: Distribution of slur scores in the raw dataset.

As a result of the above steps, for each song lyrics, we extracted profanities
and matched them with a list of profanity categories from The Obscenity List
777We should note that the set of profanities is limited by The Obscenity
List.. The distribution of profanities in the filtered dataset is shown in
Figure 4.

Figure 4: Distribution of profanities across matched categories in the
filtered dataset.

At the end of these steps, we obtained two corpora:

  * •

$Slurs$ corpus, which is the corpus we collected, cleaned, and annotated; it
comprises 15,282 songs (6,514,188 tokens).

  * •

$Mitislurs$ corpus, composed of 10,748 songs (4,531,673 tokens) which
represents a subset of the Slurs corpus, that has been filtered to mitigate
the occurrence of profanities (as defined in Section 5).

##  6 Raply

In this section, we describe Raply – an automatic rap generation system.

###  6.1 Model Overview

We used a GPT-2 Radford et al. (2020) model, which is a generative language
model based on Transformers decoder. Training such a model comes in two steps:
an unsupervised pre-training on a large quantity of data followed by a
supervised fine-tuning step. In the case of an auto-completion task, the input
of the model is the concatenation of the query $p$ and its extension $q$, with
a delimiter token in between. Since our goal is to explore the effectiveness
of GPT-2 on generating rap, instead of training the model from scratch, we use
a pre-trained GPT-2 model and fine-tune it on our corpora for the task of rap
lyrics generation.

###  6.2 Experiments

We run the experiments on the two built corpora. Due to the nature of input
data required for fine-tuning GPT-2 model, we concatenate verse lines,
separated by the special token ’Line:’. The pre-trained model we used was
trained on BookCorpus Yao and Huang (2018), a dataset of over 7,000
unpublished fiction books from various genres. We reuse the byte-pair encoding
vocabulary of this model.

We randomly select 10% examples from data as the test set and use the
remaining examples as the training set. For both corpora, we use Adam
optimizer with a learning rate of $1e^{-5}$ and a batch size of 32. We train
the models on 3 epochs. The training phase was done in both
Kaggle888https://www.kaggle.com/. platform as well as Grid
5000999https://www.grid5000.fr/. on Nvidia GPU P100.

After the training phase, we proceed with the generation phase. We split each
instance of the test set into two parts, an input part, and a reference part,
the input is fed to the model during the generation and the reference is used
as a baseline. Our code is made available on GitHub
101010https://github.com/ManilBen/Raply..

We request the model to generate sequences using top-$k$ sampling method Fan
et al. (2018), the value of $k$ is set to 50 and the minimum and maximum
length of the output sequence to 4 and 50 respectively.

For each generated sequence, we compute its rhyme density, a metric introduced
by Malmi et al. (2016) to evaluate the quality of rhyming fluency. Rhyme
density has shown a correlation with a human evaluation of rap lyrics’
technical quality, which makes it a suitable metric for evaluating the models.
A rhyme density greater than 1 can be considered as high. It should be taken
into account though, that this metric only considers one aspect of rap, namely
rhyme, and ignores other crucial components of rap like figurative language,
the context, the meaning of the words, etc.

Since rhyming is not always identifiable through orthography, we use the
program eSpeak111111https://espeak.sourceforge.net/. to transform each token
into its phoneme form in the IPA 121212International Phonetic Alphabet. and
then find the longest matching vowel sequence near each token. We then average
the length of the sequence overall words to obtain the value of rhyme density.
This process is applied to both the generated sequence and the reference part
of the test set. We choose to mainly focus on the rhyming aspect. We justify
this by the fact that GPT-2 model already showcases satisfying semantic
preservation abilities and that rap lyrics are inherently not strict when it
comes to semantic preservation in comparison with poetry for example. We also
measure the slur score on the generated text as defined in the formula defined
in Section 5.

###  6.3 Results and discussion

We compare the performances of the GPT-2 model on our two corpora ($Slurs$ vs
$Mitislurs$) and compare the perplexity and rhyme density values of the
generated text with the human-generated one from our test set. Perplexity is
frequently used to assess the quality of language models, and measures how
well a probabilistic model is able to predict a sample. A lower perplexity
means that the model is more confident in its predictions and, as a result, is
less startled by the data.

As Table 1 shows, the model GPT-$2_{Mitislurs}$ is able to improve the rhyme
density and outperforms Ghostwriter 131313The value is adopted from Xue et al.
(2021). Potash et al. (2015). However, we are not able to outperform Dope
Learning model Malmi et al. (2015) in terms of rhyme density. We explain this
difference by the fact that Dope Learning model does not properly generates
new lines from scratch but rather predicts the next line of rap from an
already rapper-written sample of lines.

Model |  Reference RD |  Generated RD |  PP
---|---|---|---
Ghostwriter |  - |  0.17 |  -
GPT-$2_{Slurs}$ |  0.6 |  0.7 |  41.6
GPT-$2_{Mitislurs}$ |  0.65 |  0.8 |  46.3
DopeLearning |  - |  1.4 |  -
Table 1:  Comparative results of rhyme density (RD) and perplexity (PP).

To assess if the profanity-mitigation approach had an impact on the generated
text, we evaluate the slur score of generated text by the developed model
while training on both $Slurs$ and $Mitislurs$ corpora, and for a sample from
generated data from Malmi et al. (2015), made available by the
authors141414https://github.com/ekQ/dopelearning/random_songs.zip.

Table 2 shows the average slur score values, we notice that the use of
$Mitislurs$ corpus helped lower the occurrences of profanities in the
generated text, and outperforms Dope Learning model. This does not mean that
profanities will be completely absent in the generated text, our goal is to
mitigate the presence of these words. As Table 3 shows a sample from the
generated text of GPT-$2_{Mitislurs}$, profanities can still occur.

Model | Slur score
---|---
GPT-$2_{Slurs}$ | 0.02
GPT-$2_{Mitislurs}$ | 0.008
Dope Learning | 0.23
Table 2:  Obtained values of slur score of GPT-$2_{Slurs}$ and GPT-$2_{Mitislurs}$ models. Input: |  And now I’m in the rap game, like the crack game
---|---
Output: |  I got enemies coming, I play my game
|  I can be king or queen, like my style,
|  I play like the King of Pain ”Sonic Yoda”
Input: |  I take my time when I rhyme lovely, me, nah like ugly n****
Output: |  Tapping out the trap doors and holla ’bout me
|  B***** they gonna nuke if they can break me
Table 3:  Example lines generated by the GPT-$2_{Mitislurs}$ model. Note: the highlighted text represents either rhyme or alliterations. Input: |  Takin orders from a snitch that sleeps with cops
---|---
Output: |  You n***** out to f***, my f*****, you a b**** and you some s*** yo
Input: |  Tell me that they love me, I don’t give a f****
Output: |  I wanna put on a black man’s d*** and let y’all f*** out his m*********
Table 4:  Example lines generated by the GPT-$2_{Slurs}$ model.

##  7 Further work

Pertaining to the direction of this paper, we aim to improve text generation
by implementing a ranking algorithm during the lyrics generation phase.
Therefore, we should be able to select the line which adheres the most to a
rhyming pattern. In order to achieve the aforementioned task, we also aim to
train the model on a curated corpus that follows better rhyming composition.
That is to say, the source data should have a clearer rhyming pattern that
will be able to train the model to produce the desired text output.
Additionally, the text generation will be tested using more efficient models
like GPT-3 and compared to previously generated data to compare and measure
the results obtained from previous models. Lastly, we plan to focus on
qualitative analysis of model results.

##  8 Conclusion

In this paper, we present Raply, a GPT-2-based model for rap generation, which
takes into account offensive content and tries to mitigate its occurrence. We
achieved it through the collection of a more ethical corpus $Mitilslurs$ and
thus we reduced the presence of offensive content in the generated rap. The
developed model is also able to generate close-to-human-like rhyming patterns
according to the rhyme density metric.

To our knowledge, Raply is the first model that tackles the issue of offensive
content in the context of rap lyrics generation.

##  9 Environmental impact

We do care about the environmental effects of our NLP models on energy
consumption and carbon emissions. Below we provide the information considering
our models’ training. Assuming that in order to train each of the models we
applied the maximum GPU power, i.e. 250 watts, then the energy consumed can be
obtained using the following formula:

$E=Pt$

We have therefore calculated the energy consumption for each trained model per
day (Table 5). In total, therefore, we spent 3,25 kilowatts hour per day on
training the models. For comparison, we have borrowed the energy consumption
statistics for 2019 from the EIA151515https://www.eia.gov/. website. Thus,
each resident in France consumes 17,6 kilowatts of energy per day, which is 5
times as much as our trained models.

Model | Training hrs/Day |  kWh/Day
---|---|---
GPT-$2_{Slurs}$ | 6 |  1.5
GPT-$2_{Mitislurs}$ | 7 |  1.75
Total | 13 |  3.25
Table 5:  Energy consumption of GPT-$2_{Slurs}$ and GPT-$2_{Mitislurs}$ per
day.

##  10 Ethical considerations

Working with text generation models is challenging in the sense that the
trained model is a kind of ”black box” which is difficult to supervise. This
raises significant ethical issues that must be addressed. As we deal with rap
lyrics generation, we anyway encounter ethical issues, in particular,
discrimination, as the focus of our paper is shifted to handling profanity
categories. This paper contains examples of words that may be abusive to
certain categories of people. Unfortunately, the occurrence of slurs in rap
lyrics is virtually inevitable, since the presence of profanities is part of
rap culture. However, in this paper, we aim to reduce the number of slurs in
the generated data. It is also essential to stress that the examples of
generated lines containing slurs by no means reflect the attitude of the
authors toward any categories of people. There is no way we can affect models
already trained on enormous amounts of data, but we attempt to retrain the
model on our own data to somehow mitigate occurring slurs and obtain less
offensive verses.

Another critical concern that we have to deal with is the bias of our model.
Since the Obscenity list consists of unbalanced categories of slurs, and
certain categories of slurs (e.g. sexual anatomy/acts) tend to occur in the
training data, there is a risk that the model will generate more slurs of the
same category. Consequently, one should attempt to mitigate the bias, for
example, by balancing the songs in the training lyrics dataset according to
the presence of profanities of the particular category, or by trying to
control the number of slurs directly during generation.

A further matter of no small significance is that of legal issues. The
possibility of user-generated content on websites like Genius violating
intellectual property rights (IPR) also needs to be addressed, as Genius
disclaims any responsibility for any content posted by users. Because it could
be difficult or even impossible for others to utilize the data or build upon
it if it is not legally permitted to do so, this can pose issues for
reproducibility. It is crucial that Genius, or any other platform, take
proactive steps to address these issues and establish a clear, consistent
policy for handling user-generated content that can violate IPR.

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##  Appendix A Appendix

In Table A are shown some examples of offensive language related to each
profanity group:

Profanity group | Examples
---|---
sexual anatomy / sexual acts | ass kisser, dick, pigfucker
bodily fluids / excrement | shit, cum
sexual orientation / gender | faggot, tranny, bitch, whore
racial / ethnic | chink, n3gro
mental disability | retard, dumbass
physical disability | quadriplegic bitch
physical attributes | fatass, ugly whore
animal references | pigfucker, jackass
religious offense | goddamn
political | China virus
other / general insult | mofucker

Table 6:  The examples of profanities for each group from The Obscenity List.

Input: |  Sicken n*****, what I nickname my gun? Robitussin
---|---
Output: |  I ain’t with the firin’ shit, I ain’t with the tusslin’
Input: |  Military hat, fatigue pants saggin’, Henny’d up
Output: |  Gotta play nice when it doesn’t rhyme
Input: |  Some spacious, others is basic
Output: |  It’s good we move with good habit
Input: |  All my shit is Bohchah Fam
Output: |  Oh, wait, wait! I got a n**** on tape
Input: |  I take my time when I rhyme lovely, me, nah like ugly n****
Output: |  Tapping out the trap doors and holla ’bout me
|  B***** they gonna nuke if they can break me
Input: |  One of the livest n***** in New York
Output: |  I’ve got it in their veins, his rap verse proclaims
Table 7:  Example lines generated by the GPT-$2_{Slurs}$ model.


11institutetext:  University of Pisa 22institutetext: INFN Pisa

# Status of the Fermilab Muon $g-2$ Experiment

Paolo Girotti on behalf of the Muon $g-2$ (E989) Collaboration 1122
<EMAIL_ADDRESS>
###### Abstract

The muon anomalous magnetic moment, $a_{\mu}=(g-2)/2$, is a low-energy
observable which can be both measured and computed with very high precision,
making it an excellent test of the Standard Model (SM) and a sensitive probe
of new physics. Recent efforts improved the precision of both the theoretical
prediction and the experimental measurement. On the theory side, the Muon
$g-2$ Theory Initiative, an international team of more than 130 physicists,
reached in 2020 a consensus on the SM prediction for $a_{\mu}$. On the
experimental side, the E989 Muon $g-2$ Collaboration at Fermilab (FNAL)
published in April 2021 a new measurement of $a_{\mu}$ from the Run-1 dataset
(2018) with 0.46 ppm precision, corroborating the previous Brookhaven National
Laboratory (BNL) measurement and increasing the discrepancy with the SM value
to 4.2 standard deviations. In this paper, I will discuss the experimental
setup and report on the current status of the experiment.

##  1 Introduction

The study of the lepton magnetic moments has a long history starting back in
1928 when Dirac predicted $g=2$ for the electron. The 1948 calculation by J.
Schwinger of the first-order correction due to virtual particles marked the
beginning of a long sequence of improvements in both the theoretical
prediction and the experimental measurement of the electron and muon
anomalies. While the electron $g-2$ measurements represent a striking
confirmation of QED theory, the muon anomaly has emerged as an important test
of the entire Standard Model due to significant contributions from weak and
strong interactions.
In April 2021, the Muon $g-2$ Collaboration released their Run-1 measurement
of the muon anomaly with a precision of 460 ppb PRL . This result is in
agreement with the previous experiment at Brookhaven National Laboratory
(E821), which determined $a_{\mu}$ with an accuracy of 540 ppb BNL . By
combining the two measurements into a new world average, we find a discrepancy
of 4.2 $\sigma$ with respect to the current consensus of the Standard Model
prediction WP2020 . The current theoretical and experimental values are:

$\displaystyle a_{\mu}^{Th}\;\;[2020]$ $\displaystyle=116\,591\,810(43)\times
10^{-11}\textnormal{ (0.37 ppm)}$ (1) $\displaystyle a_{\mu}^{Exp}\,[2021]$
$\displaystyle=116\,592\,061(41)\times 10^{-11}\textnormal{ (0.35 ppm)}$ (2)
$\displaystyle a_{\mu}^{Exp}-a_{\mu}^{Th}$ $\displaystyle=(251\pm 59)\times
10^{-11}\textnormal{ (4.2$\sigma$)}$ (3)

The history of $g-2$ is far from being completed, as the Muon $g-2$ Experiment
at Fermilab will improve its precision by a factor of four, other experimental
efforts are in the works, and new calculations with Lattice QCD techniques
represent a new way to improve on the theoretical predictions.

Figure 1: Run-1 FNAL measurement compared with the previous BNL experiment and
the 2020 Standard Model consensus recommended by The Muon $g-2$ Theory
Initiative WP2020 . Figure from PRL .

###  1.1 The importance of g-2

The motivation for muon $g-2$ experiments comes from various reasons. First of
all, the muon anomaly $a_{\mu}\equiv\frac{g-2}{2}$ is a relatively simple
observable to measure. The $g$ factor relates the magnetic moment of the muon
with its spin:

$\vec{\mu}=g\frac{q}{2m}\vec{S}.$ (4)

Such measurements can be obtained through a spin-polarized beam of muons, an
external and measurable magnetic field, and a technique to determine the spin
precession through time. While such a setup is easy to create with current
technology, many challenges arise when trying to achieve an accuracy of less
than 1 part-per-million (ppm).
Despite the difficulties, we pursue precision measurements of $a_{\mu}$
because the anomaly probes all of the interactions between the lepton and
virtual particles. The whole Standard Model is tested, since all possible
particles can contribute either via direct virtual interaction with the muons
or with higher-order loop corrections, adding to the value of $a_{\mu}$.
Figure 2 depicts the lower-order interactions for the electromagnetic, weak,
and strong forces. Table 1 lists the contributions to $a_{\mu}$ by the three
SM forces.

Figure 2: Lower-order examples for EM, weak, and QCD interactions PRL .

Moreover, eventual measurements that differ from the SM theoretical prediction
would indicate hints of new physics. Such results would help setting new
constraints on the multitude of currently proposed Beyond Standard Model (BSM)
theories.
Muons are particularly important in such a measurement because BSM
interactions with massive particles contribute with mass suppression terms,
$\propto(\frac{m_{lepton}}{M})^{2}$. The electron $g-2$ has already been
measured with around $\mathcal{O}(10^{3})$ times more precision than the muon
Hanneke , but the relative mass ration between the electron and the muon
enhances the sensitivity to these terms by a factor of
$(\frac{105.66}{0.511})^{2}\approx 43000$ Jegerlehner . Following the same
reasoning, $\tau$ leptons would be even better for detecting new heavy
particles, but their very short lifetime $(2.9\times 10^{-13}$ s), multiple
decay branches, and other practical reasons make such an experiment not
feasible with current technology.

Table 1: Standard model contributions to $a_{\mu}$ WP2020 . Term | Value ($\cdot 10^{-11}$)
---|---
$a_{\mu}^{QED}$ | $116\,584\,718.931\pm 0.104$
$a_{\mu}^{EW}$ | $153.6\pm 1.0$
$a_{\mu}^{HVP}$ | $6\,845\pm 40$
$a_{\mu}^{HLBL}$ | $92\pm 18$
Total SM | $116\,591\,810\pm 43$

##  2 The experiment

The new Muon $g-2$ Experiment in operation at Fermi National Accelerator
Laboratory aims to measure the muon’s anomalous magnetic moment with a
precision of 0.14 ppm, a factor of four better than the previous BNL E821
Experiment. To achieve this precision, a statistics of $\sim$20 times the
amount collected at BNL is needed, and the systematics must be contained
within 100 ppb. In order to obtain a large number of observed muons, the BNL
storage ring was moved from Brookhaven to Fermilab and installed in the FNAL
muon campus accelerator chain, where the beam is cleaner and much more
intense.
The experimental technique consists of producing a polarized and clean beam of
muons, sending it to a storage ring with very uniform magnetic field, and
observing the decay positrons. Three measurements are needed to calculate the
muon anomaly $a_{\mu}$: the muon anomalous precession frequency, the magnetic
field intensity, and the beam distribution inside the storage region. The
master formula is the following:

$a_{\mu}=\frac{\omega_{a}}{\tilde{\omega}_{p}}\frac{g_{e}}{2}\frac{m_{\mu}}{m_{e}}\frac{\mu_{p}}{\mu_{e}}\leavevmode\nobreak\
,$ (5)

where $\omega_{a}$ is the muon anomalous precession frequency, and
$\tilde{\omega}_{p}$ is the Larmor precession frequency of the proton
($\omega_{p}$) convoluted with the beam distribution, representing the average
field intensity experienced by the muons. The remaining factors are known with
sufficient precision from other experiments.

###  2.1 The muon beam

The muons observed in the Muon $g-2$ Experiment originate from decaying pions,
which are in turn produced by the collisions of an 8 GeV proton beam on an
Inconel® 600 target. After the collisions, secondary particles are focused
with an electrostatic lithium lens, and pions having 3.1 GeV momentum are
extracted. The beam of pions then circulates inside a Delivery Ring, where
remaining protons get removed and all the pions decay into muons. Since pions
have zero spin, the muons are emitted isotropically in the rest frame of
reference, but their helicity is constrained by the weak decay, as illustrated
in Fig. 3. By selecting boosted muons with higher momentum, it is possible to
obtain a highly polarized beam in the laboratory frame of reference.

Figure 3: Decay of the positive pion in the rest frame of reference. The spin
of the muon points toward the pion as the neutrino must be left-handed because
of parity violation of the weak interaction.

Finally, the 120-ns long muon bunches enter the Muon $g-2$ storage ring at an
instantaneous rate of 100 Hz (11.4 Hz average). A superconducting inflector
magnet lets the beam pass through the ring yoke into the storage region. Then,
after the first quarter of an orbit, a fast pulsed kicker magnet deflects the
muon bunch into the final orbit, which has a radius of 7.112 m and a period of
149.2 ns. The $\sim$5000 stored muons per bunch then circulate inside the ring
for 700 $\mu$s, while a set of electrostatic quadrupoles provides weak
focusing for vertical confinement.

###  2.2 The field measurement

The magnet field measurement consists of various components and techniques
used to determine the value of $\omega_{p}$: 378 Nuclear Magnetic Resonance
(NMR) fixed probes are placed along the ring under and over the vacuum
chamber. The probes keep monitoring the field during the whole data taking.
Every three days during data taking, trolley runs are performed in which a
mobile cylinder equipped with 17 NMR probes traverses the beam storage region
to make 9000 measurements along the ring Field . The trolley measurements
produce a three-dimensional (3D) map of the magnetic field inside the storage
region. The 3D map is then interpolated over the time between two consecutive
trolley runs with the fixed probes data. While the magnetic field is ppm-level
uniform along the beam cross section, it is necessary to know the muon
distribution inside the storage ring in order to achieve an uncertainty of 70
ppb on $\tilde{\omega}_{p}$. For this, a set of two tracker stations observe
positron tracks and extrapolate the decay vertex in the storage orbit. The
beam distribution is extrapolated over the rest of the ring with data from the
calorimeters and multiple dedicated simulations Beam .

###  2.3 Muon precession frequency measurement

A charged particle with mass $m$ placed in an uniform external magnetic field
will follow a circular path because of the the Lorentz force, and this motion
is called cyclotron motion. If the particle has spin, the spin direction will
also rotate (precess) around the direction of the magnetic field. In the
absence of electrical fields, and with the particle velocity and spin
perpendicular to the magnetic field, the equations governing this motion are:

$\displaystyle\vec{\omega}_{s}$
$\displaystyle=-\frac{ge\vec{B}}{2m}-(1-\gamma)\frac{e\vec{B}}{m\gamma}$ (6)
$\displaystyle\vec{\omega}_{c}$
$\displaystyle=-\frac{e\vec{B}}{m\gamma}\leavevmode\nobreak\ ,$ (7)

where $\omega_{s}$ is the spin precession angular frequency and $\omega_{c}$
is the cyclotron frequency. The second term of Eq. 6 is the Thomas correction
to the Lorentz force, which accounts for the rotation of the particle’s frame
of reference. Subtracting Eq. 7 from Eq. 6, we obtain the rotation of the
particle spin with respect to its momentum. This is the anomalous precession
frequency $\omega_{a}$:

$\vec{\omega}_{a}\equiv\vec{\omega}_{s}-\vec{\omega}_{c}\simeq-\left(\frac{g-2}{2}\right)\frac{e\vec{B}}{m}\equiv-
a_{\mu}\frac{e\vec{B}}{m}\leavevmode\nobreak\ .$ (8)

However, the particle beam is not always perfectly parallel to the storage
plane, but oscillates and breathes vertically and horizontally. The anomalous
precession frequency is sensible to such effects:

$\displaystyle\vec{\omega}_{a}=$
$\displaystyle-\frac{e}{m}\Bigg{[}a_{\mu}\vec{B}-\left(a_{\mu}-\frac{1}{\gamma^{2}-1}\right)\frac{\vec{\beta}\times\vec{E}}{c}$
$\displaystyle-
a_{\mu}\left(\frac{\gamma}{\gamma+1}\right)(\vec{\beta}\cdot\vec{B})\leavevmode\nobreak\
\vec{\beta}\leavevmode\nobreak\ \Bigg{]}\leavevmode\nobreak\ .$ (9)

The storage region houses several electrostatic quadrupoles for a weak
vertical focusing of the beam. This would affect the muon precession
frequency, but, for a muon beam with magic momentum $p_{\mu}$ = 3.094 GeV/c,
corresponding to a value of $\gamma=29.3$, the second term of Eq. 9 cancels
out. The remaining effect and the third term are measured separately and
applied as E-field and Pitch corrections to $\omega_{a}$.

The measurement principle relies on the parity-violating nature of the weak
decay. The positive muons decay into a positron and two neutrinos with nearly
$100\%$ probability. In the rest frame of the muon, the highest energy decay
positrons come from decays in which the neutrinos are emitted collaterally, as
depicted in Fig. 4. In this scenario, half of the initial rest mass of the
muon is carried away by the decay positron ($E_{max}\approx 53$ MeV), while
the rest is shared by the two neutrinos. Since the neutrino and anti-neutrino
are traveling in the same direction, and the weak decay dictates they must
have opposite helicities, their spins must be opposite. With the neutrinos’
spins canceling, conservation of angular momentum forces the decay positron to
carry the spin of the parent muon.

Figure 4: Muon decay configuration that maximizes the positron momentum and
decay probability. The right-handed positron is emitted in the direction of
the muon spin.

The $V-A$ nature of the weak decay prefers to couple to a right-handed
positron, so the high-energy decay positron depicted in Fig. 4 tends to be
emitted in the direction of the muon spin. Therefore, in the rest frame of the
muon, the spin direction of the muon can be monitored by observing the
direction at which the high energy decay positrons are emitted. The Muon $g-2$
Experiment is equipped with 24 electromagnetic calorimeters located around the
ring with 15Â azimuthal distance between each other. The asymmetry of the
decay process, together with the fact that the spin precesses with respect to
the momentum, results in an oscillation in the count of positrons over time.
The number of detected positrons above a single energy threshold $E_{th}$ is:

$N(t)=N_{0}e^{-t/\tau}[1+A\cos(\omega_{a}t+\phi)]\leavevmode\nobreak\ ,$ (10)

where the normalization $N_{0}$, the asymmetry $A$ and the initial phase
$\phi$ are all dependent on the energy threshold. $\tau$ represents the
lifetime of the muon in the laboratory frame of reference, that is
$\gamma\tau_{\mu}\simeq 64.4$ $\mu$s. Additional beam-related effects, as
coherent betatron oscillations and muon losses, appear as multiplicative terms
to $N_{0}$, $A$, and $\phi$. Three different reconstruction techniques and
several analysis procedures are implemented to obtain the $\omega_{a}$ value
Omega . As an example, in the A-Weighted method the positrons counts are
weighted by the asymmetry, which depends on the energy, yielding the maximum
possible statistical power for a given threshold $E_{th}$. The typical plot
showing the count of positrons over time is called wiggle plot and is shown in
Fig. 5.

Figure 5: Wiggle plot showing the number of $>$1.7 GeV positrons over time.
The graph is wrapped every 100 $\mu$s to show the entire measurement period.
The oscillation period is $2\pi/\omega_{a}$.

The positrons are measured by 24 calorimeters, which each have a 9x6 array of
PbF2 crystals with size $2.5\times 2.5\times 14$ cm. The charged particles in
the electromagnetic shower generate Čerenkov photons. The choice of a pure
Čerenkov material is driven by the almost instantaneous signal produced when a
positron strikes, further enhanced by a black Tedlar® wrapping that absorbs
part of the reflected photons. Every crystal is coupled with a Silicon
PhotoMultiplier (SiPM) detector, whose signals are digitized at an 800 MHz
sampling rate and chopped into 40-ns islands by on-line GPUs. A precise gain
calibration of the SiPMs at the level of $\mathcal{O}(10^{-4})$ is provided by
a Laser-based Calibration System Laser .

###  2.4 Blinding

The Muon $g-2$ Experiment has an unique feature, which is the hardware
blinding of the digitization clocks. Before the start of each run of data
acquisition, the main clock that determines the calorimeters digitization rate
is configured with a secret offset in the 25 ppm range. The real frequency is
known only by two persons outside of the collaboration, and that number is
only revealed to the analyzers at the final stage of the measurement process.
The whole muon precession analysis is therefore performed blindly on time-
uncalibrated data. When the collaboration unanimously decides that the
analysis is mature and the final value is calculated, the secret frequency is
revealed.

##  3 Status and future of the Muon $g-2$ Experiment

The Muon $g-2$ Experiment at Fermilab is currently running its fifth year of
data taking. In the first four years, an equivalent statistics of
13$\times$BNL has been collected (Fig. 6). The Run-1 data has been completely
analyzed and has been published earlier this year. The result was in agreement
with the BNL experiment and the combined value increased the discrepancy with
theory to 4.2 $\sigma$ PRL .
The data from Run-2 through Run-4 data is currently under reconstruction and
analysis, with many efforts in improving the systematic errors with respect to
Run-1, both on the hardware and the software fronts. For instance, the magnet
has been covered with a thermal blanket, and the hall is now temperature-
controlled with air conditioners. The kicker strength has been improved Kicker
, and the electrostatic quadrupole resistors have been upgraded. New beam
scraping techniques and quadrupole radio-frequency pulses have been put in
place to increase the storage efficiency and minimize the betatron
oscillations QuadRF . On the analysis side, independent efforts from the
$\omega_{a}$ groups try to increase pileup separation and decrease gain
related systematics. The beam-related effects to $\omega_{a}$ are being
calculated with improved precision thanks to ever increasing simulation
studies. The Run-5 data taking just started in late 2021, and another Run-6 is
currently planned. The collaboration is considering the possibility of using
negative muons for the Run-6 data taking. Such a run would provide an
opportunity to test the robustness of the method and verify CPT and Lorentz
conservation.

Figure 6: Collected statistics compared to the BNL dataset in the first four
runs of the experiment.

##  4 Acknowledgments

This work was supported in part by the US DOE, Fermilab, the Istituto
Nazionale di Fisica Nucleare (Italy), and the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreements No. 690835 (MUSE) and No. 734303 (NEWS).

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