# Increased phase precision of spatial light modulators using irrational
slopes: Application to attosecond metrology

Geoffrey R. Harrison  Tobias Saule  Brandin Davis  Carlos A Trallero-Herrero
Department of Physics, University of Connecticut, Storrs, Connecticut 06269,
USA<EMAIL_ADDRESS>
(May 2022)

###### Abstract

The ability of spatial light modulators (SLMs) to modify the amplitude and
phase of light has proved them invaluable to the optics and photonics
community. In many applications the bit-depth of SLMs is a major limiting
factor dictated by the digital processor. As a result, there is usually a
compromise between refresh speed and bit-depth. Here we present a method to
increase the effective bit-depth of SLMs which utilizes a linear slope as is
commonly applied to deal with the zeroth order effect. This technique was
tested using two interferometric transient absorption spectroscopy setups.
Through the high harmonic generation in gases producing a train of attosecond
pulses and harmonics from solids in the ultraviolet, two pulses are generated
that interfere in the far field providing a measurement of the optical phase.
An increase in the precision far beyond the limit dictated by the digital
processor in the bit-depth was found.

††journal: ao††articletype: Research Article

##  1 Introduction

For the past few decades spatial light modulators (SLMs) have found widespread
use in the optics and photonics community because of their ability to readily
manipulate the properties of light, e.g. amplitude and phase. They are used
extensively in applications as diverse as optical microscopy[1, 2], imaging
though scattering media[3, 4, 5], material processing[6, 7], quantum
entanglement experiments[8, 9, 10], 3-D displays[11, 12, 13], optical
tweezers[14, 15, 16], beam shaping[17, 18], and even everyday applications
like data storage[19, 20]. While there are many advantages to them, most SLMs
have common limitations that are inherent to the digital technology used to
drive electro-optical changes in the modulator. Thus, besides the number of
pixels which determines the spatial optical resolution, the pixel refresh
speed and phase resolution are completely determined by the bit rate of the
digital infrastructure. Other known issues present in SLMs such as spatial
phase variance and non-linear phase response can be mitigated by time
consuming calibration steps in the form of a wavefront correction mask and
phase look up table respectively. Ultimately, however, the bit-depth and
spatial resolution, which are intrinsic properties of the device, limit how
precisely and with what spatial resolution a phase can be adjusted. One
application that is significantly impacted by the bit-depth is the temporal
control of attosecond pulses [21] and its application to interferometric
transient absorption spectroscopy (ITAS) as described in [22]. ITAS systems
offer some of the most precise phase measurements, with sub-attosecond
precision, by controlling the optical phase of the fundamental which
translates into a time delay/advance of a train of attosecond pulses. However
the bit-depth defines the minimum time resolution, i.e. time step size, that
can be measured. As an example, for an 800 nm laser incident on an 8 bit-depth
SLM with a $2\pi$ phase range the minimum temporal step size is
$T_{800}/2^{8}\approx 10$ as (as = attosecond = $10^{-18}$s), with $T_{800}$
the period of the 800 nm pulse. As ultrafast physics moves into the realm of
zeptoseconds (zs = $10^{-21}$s) achieving sub-attosecond time steps is clearly
desirable [23]. As an example, to sample dynamics with sub 100 zs steps would
require an SLM with over 16 bit-depth which is not readily available.

Another limitation of SLMs is the zeroth-order beam [24] which co-propagates
with the desired light but is unaffected by the SLM. While some applications
can tolerate this issue, others like holographic data storage[25] and ITAS,
need to address and suppress this effect to improve performance. The simplest
and most common way to correct for zeroth order is to superimpose a linear
phase slope on top of the desired phase mask, tilting the beam away from its
original direction and separating the two beams in the far field[26]. This
linear slope has just recently been shown to have another benefit, namely a
reduction in temporal phase fluctuations[25] which is predicted to improve the
quality of holographic data storage systems.

Here we present a method to mitigate the bit-depth limit and increase the
phase precision of SLM devices. Similar to phase fluctuation suppression[25]
this technique is based on the addition of irrational linear slopes in
addition to a desired phase. That leads to an increase in the effective bit-
depth of the device via an effect similar to volume averaging. This method is
verified using precise ITAS setups where linear slopes, combined with an
overall phase offset were used to achieve step sizes in the time delay of
attosecond pulses 3-6 times smaller than what would be achievable otherwise.
We believe this method could be used to gain multiple orders of magnitude more
precision, beyond what our system could measure. This technique was tested
with SLMs from two major producers, Meadowlark and Hamamatsu, to demonstrate
the generality of its use.

##  2 Experimental setup

To test this technique we measured the phase applied by two SLMs using a self-
referencing interferometer for attosecond pulse trains as described in [21].
In short, two intertwined linear slopes create two identical foci after a
lens. Each focus generates high harmonics (HHG) in gases [27, 28] or lower
harmonics in solids [29] employing different setups. The experimental setup
for the gas experiments is shown in Fig.1(a). 790 nm, 2.5 mJ, 35 fs pulses
from our titanium-sapphire laser were separated by a XY Phase Series 512L
Meadowlark SLM into two beams that are focused through a 300 mm lens into two
foci in an ethylene gas jet. Since the masks for the two foci already use
linear slopes no zeroth order offset was used for these experiments. The SLMs
used in these experiments are anti-reflex coated so the zeroth order is
minimal and is not intense enough to generate the harmonics examined here.
This SLM has a resolution of 1920x1152 pixels covering an area of 17.6 by 10.6
mm and an 8 bit-depth. Harmonics are generated at both foci and are sent into
a spectrometer where they interfere spectrally (still yielding discrete
harmonics) and spatially creating fringes in each harmonic (Fig.1(b)). The
spectrometer uses a laminar, curved, variable spacing extreme ultraviolet
(XUV) grating, followed by a micro channel plate phosphor stack and a low
noise camera (Hammamatsu Orca Flash). The phase difference between the two XUV
beams, shown in Fig.1(c), can be controlled through the SLM and measured
directly from the interference fringes. A phase offset $\Delta\phi$ given to
one of the fundamental beams translates into a phase advance of the harmonic
$q$ given by $q\Delta\phi$. The phase added by SLMs is often described in
units of radians, however, when comparing phase masks of different SLMs the
most natural unit to use is bits. Using bits allows us to talk about effects
that apply to all SLMs regardless of the central wavelength of design, lookup
tables etc. Conversion to radians from bits and vice versa is done by noting
that $2\pi$ rads of the central wavelength of operation is usually equal to
$2^{\mbox{bit-depth}}$ bits.

Experiments were repeated using a Hamamatsu X10468 SLM which has a resolution
of 792x600 pixels covering an area of 15.8 by 12 mm with 8 bit-depth. Since
this SLM operates at a different wavelength, 1400 nm, a separate but similar
setup, working with solids, was used to test it (Fig. 1 (d)). In this setup
790 nm pulses from our laser are sent into an optical parametric amplifier
(OPA) where they generate 100 $\mu$J, 55 fs 1400 nm pulses. Similar to before,
these pulses are split by the Hamamatsu SLM and then focused with a calcium
fluoride lens onto a quartz sample where harmonics are generated in the
reflection geometry[30]. Harmonics are spectrally separated by a flat grating
and the interference fringes for each harmonic are recorded by a Mightex
camera.

Figure 1:  (a) Sketch of the experimental setup using the Meadowlark SLM. A
SLM is used to separate the incoming beam into two phase-locked pulses. Both
pulses then generate spatially separated but phase-locked harmonics which
interfere in the far field and are detected in a spectrometer. (b) A sample
image from the gas spectrometer showing the $5^{th}$ to $11^{th}$ harmonics
with fringes. An integrated lineout of the $9^{th}$ harmonic, outlined in red,
is shown below. (c) A phase offset in one focus leads to a difference between
both fundamental foci which in turns creates a phase difference between the
harmonics. (d) Sketch of the experimental setup used for phase measurements in
solids using a Hamamatsu SLM. A small portion of one of the signals from an
OPA is sent onto the SLM at a steep angle. The diffracted beam is then focused
with a calcium fluoride lens onto the quartz sample where harmonics are
generated in the reflection geometry. Harmonics are collimated with a
cylindrical mirror and sent onto the camera where spectral fringes are
recorded.

Both setups use similar masks to split the incoming beam into two foci and
then add a phase offset to one of those foci. Because of the unambiguous
relation between phase offset and delay of the harmonics we will refer
throughout the paper to phase offsets and phase delays indistinguishably.

##  3 Limits of SLM bit-depth

The masks used in these experiments were made of two intertwined linear slopes
each applied across the entire SLM. The linear slopes split the beam into two
angled wavefronts which focus 300 $\mu m$ away from each other after being
focused by a lens. As can be seen in Fig. 2(a) the masks are made by randomly
assigning each pixel to one of the slopes. These random masks are used instead
of other patterns, such as checkerboards since they have better diffraction
efficiency. As expected, these masks add some dispersion because the SLM
applies slightly different slopes to different wavelengths due to the
wavelength dependent phase of each pixel. However, we find this dispersion to
be minimal as quantified by a frequency resolved optical gating (FROG)[31]
shown in Fig. 2(b) measured from a beam split by the SLM as described above.
The pulses are stretched from 35 fs to 39 fs remaining short enough to
generate harmonics for ITAS. Phase delays between the two beams can be created
by adding a constant offset to every pixel assigned to one of the slopes. This
delay is multiplied in the harmonic generation and can then be measured from
the interference in the spectrometer.

Figure 2:  (a) A simplified schematic of how the SLM masks used in these
experiments are generated. The slopes have equal but opposite angles and each
has half of the pixels randomly assigned to it. (b) A FROG trace and
autocorrelation of the pulses after being split by the SLM. They show an
autocorrelation full width at half maximum, $\tau_{AC}$, of 54 fs
corresponding to a gaussian pulse width, $\tau_{Gauss}$, of 39 fs.

Although up to this point the masks have been described in a non-discrete
manner, the limited bit-depth of SLMs requires the pixel values to be rounded
to the nearest bit before they can be displayed on the SLM. This rounding can
eliminate information which leads to a limit for the minimum phase delay, 1
bit. This is the minimum phase that can be added to each pixel and corresponds
to $\frac{2\pi}{2^{8}}$ rad = 10.4 as in the case of our 790 nm Meadowlark SLM
with a cycle time of 2.6 fs and 18 as for our Hamamatsu SLM at 1400 nm. Figure
3 shows a measurement of the 790 nm beam’s phase as the phase delay of one
beam is scanned. For these masks the linear slopes were rounded to the nearest
bit before having the phase delay added. The delay was added in steps of
0.0249 rads of the $9^{th}$ harmonic = 0.34 bits for the Meadowlark and 0.0126
rads of the $3^{rd}$ harmonic = 0.17 bits for the Hamamatsu. Clear steps form
showing phase jumps of 1 bit for both the Meadowlark (a) and the Hamamatsu (b)
SLMs.

Figure 3:  (a) The observed phases of the $9^{th}$ harmonic generated in
ethylene as the phase delay of one beam is scanned. The red line shows how
much this harmonic would be effected by 1 bit of added phase. There are jumps
of one bit twice after every four steps. This odd jumping pattern happens
because the step size chosen hit every 5th point very well and then round to
the other bit for the rest of the cycle. (b) The observed phase of harmonic 3
generated in quartz as the phase delay of one beam is scanned. A jump of about
1 bit happens every 4 steps

##  4 Overcoming bit-depth limitations

To create phase effects of less than 1 bit the phases of multiple pixels must
be averaged together. This is accomplished by taking advantage of the rounding
that was already being done in the mask making process. This also relies on a
linear slope always being applied to the SLM to create a zeroth order offset,
on top of any other effects desired. If a phase effect is added to the mask
before it’s rounded, it can interact with the fractional part of the slope and
cause some pixels to be rounded up instead of down. The fractional part of a
number x, $\\{x\\}$, is defined as

$\\{x\\}\equiv x-\mbox{floor}(x).$ (1)

The non-fractional part of the phase will add exactly to each pixel so here
we’ll only consider phase additions of less than 1. These could be the
fractional part of a larger phase effect, separated out to simplify the math.
Assuming the fractional parts of the slope are uniformly and densely
distributed in [0,1) then the percent of pixels that flip and round up are
equal to the phase added, $d$. There are two mechanisms for a phase addition
to make a pixel round up. First if the pixel’s original fractional value is
below $\frac{1}{2}$ then $d$ can increase it above $\frac{1}{2}$ and cause it
to round up to one. Secondly, if the original value is above $\frac{1}{2}$
then $d$ can put it above $1\frac{1}{2}$ causing it to round to two instead of
one. For a pixel value, $u$, the chances for its fractional part to be above
or below $\frac{1}{2}$ are 50%. If $\\{u\\}<\frac{1}{2}$ and
$d\geq\frac{1}{2}$ then the chances of rounding up are 100%. In this case if
$d<\frac{1}{2}$ then the probability of a round up is

$P\left(\\{u\\}+d\geq\frac{1}{2}\right)=\frac{d}{1/2}=2d.$ (2)

For the second case, where $\\{u\\}\geq\frac{1}{2}$, if $d<\frac{1}{2}$ then
there is no chance of rounding up to two, otherwise the probability is

$P\left(\\{u\\}+d\geq 1\frac{1}{2}\right)=\frac{d-\frac{1}{2}}{1/2}.$ (3)

If we add together the probabilities of the two mechanisms when
$d<\frac{1}{2}$ we get

$P(\mbox{pixel
increase})=\left(\frac{1}{2}\right)\cdot(2d)+\left(\frac{1}{2}\right)\cdot(0)=d.$
(4)

Adding the probabilities when $d\geq\frac{1}{2}$ gives

$P(\mbox{pixel
increase})=\left(\frac{1}{2}\right)\cdot(1)+\left(\frac{1}{2}\right)\cdot\left(\frac{d-\frac{1}{2}}{1/2}\right)=d.$
(5)

So in the end, no matter the value of $d$, the probability of a round up
increasing the pixel value is $d$, assuming that the fractional parts of the
slope are uniformly distributed in [0,1).

The assumption of uniformly and densely distributed fractional parts is very
commonly true for SLM masks because the slopes will frequently be irrational.
Slopes are commonly defined in radians and then converted to bits bringing in
a factor of $\pi$ with the conversion, making them irrational. If the slope is
irrational then the slopes values will be multiples of an irrational number
and multiples of irrational numbers are uniformly distributed and densely
packed in [0,1) [32]. This results in an average phase equal to the desired
phase, even if that effect is less than one bit. As a simplified example take
a mask with a slope of $1\frac{1}{2}$ bits per pixel, as shown in Fig.4(a),
then $\frac{1}{2}$ a bit can be added to it without changing the slope of the
mask, as in Fig.4(b). Figure 4(c) shows the difference between the two masks;
half the pixels have been changed by one resulting in an overall phase delay
of $\frac{1}{2}$, as desired.

Figure 4:  (a) A simple example phase mask. It is a single slope angled at
$1\frac{1}{2}$ bits per pixel. All values are rounded as they would be in a
real use case. (b) A mask with the same slope but with $\frac{1}{2}$ bits
added before rounding. (c) The difference between the two masks. Every other
line has its value changed by 1 resulting in a average change of $\frac{1}{2}$
bits, as desired.

This shows that SLMs can add phase delays smaller than 1 bit but doesn’t
address the smallest delay that is achievable. There is a limit based on the
slope applied, highlighted in Fig.4 which can apply a phase delay of
$\frac{1}{2}$ bits but could not apply a phase of $\frac{1}{10}$ bits. The
slope will take values that are multiples of $1\frac{1}{2}$ and will only have
fractional values of 0 or $\frac{1}{2}$. Any addition less than $\frac{1}{2}$
will have no effect after rounding. A slope needs to reach fractional values
where the phase can change the way the slope is rounded. This limit is,
however, reduced significantly for most real-world slopes which, as discussed
before, will frequently be irrational and have fractional parts that are
uniformly and densely packed in [0,1).

There is also a hard lower limit imposed on the averaging effect; at least one
pixel must increase its value. However this limit is 1 bit divided by the
number of pixels which is extremely small as most SLMs will have somewhere
from tens of thousands to millions of pixels. The effect is worse for some
masks, like those in Fig.4, which only have a slope in one direction. For
these masks the columns all have the same values, tying them together. The
limit is then 1 bit divided by the number of different values in the slope.
These masks can be avoided by adding a second slope in the other direction,
creating a further zeroth order offset. Now each pixel has a distinct value,
although it should be noted that this second slope should depend on another
irrational number, not just $\pi$. That guarantees that no pixel values will
repeat across the SLM.

A more relevant limit, related to the previous, is the fact that the real-
valued phase must be approximated by a fraction of the pixels. Small phase
effects, closer to the one pixel limit, will be poorly approximated as such.
For arbitrary phases to be hit accurately many pixels must be flipped so that
the fraction of pixels flipped over total number of pixels is a good
approximation of the phase. This is especially important for uses such as ITAS
where many phase delay steps will be scanned through in sequence. If the
desired phase is small, and poorly approximated, each step can vary wildly as
it is either rounded up or down to the nearest phase that can actually be
displayed by the SLM. For these purposes it is very important that each step
is accurately approximated by a fraction of pixels and the easiest way to
achieve this is by having a large number of pixels flip every step. This error
will always be on the order of $\pm$one pixel flipping, which as mentioned
earlier is a small effect.

Figure 5:  (a) The observed phases of harmonic 9 generated in Ethylene as the
phase delay of one beam is scanned using the Meadowlark SLM. They are now
smooth lines with a steps size of 0.0249 rads of the 9${}^{\mbox{th}}$
harmonic = 0.34 bits. The red line shows what phase, in radians of the
harmonic, the harmonic would gain from a 1 bit addition on the SLM. (b) The
observed phase of harmonic 3 generated in quartz as the phase delay of one
beam is scanned using the Hamamatsu SLM. This is a smooth line with a step
size of 0.0126 rads of the 3${}^{\mbox{rd}}$ harmonic = 0.17 bits.

To get more specific, masks were generated with a slope of $0.124$ rads per
pixel (approximately $20\pi$ in total) in one direction and $\sqrt{2}/300$
rads per pixel in the other which are characteristic numbers for our ITAS
setup. For a typical 8-bit SLM with 500x500 pixels we conservatively estimate
that light is only incident on 300x300 of them, resulting in a minimum step
size of 0.11 zs for 800 nm light if one pixel changes. If 500 of these masks
are generated in sequence each with an additional phase delay of $0.001\mbox{
bits}=10\mbox{ zs}$, which is approximately 100 pixels flipping, then the
masks have an average delay step of $10\mbox{ zs}\pm 0.27\mbox{ zs}$, an error
of about 2.5 pixels. While the exact error of these phases varies based on
what specific phase delay is desired and what slopes are applied the error is
always similar and on the order of a few pixels flipping incorrectly. This SLM
could reliably take steps of one thousandth of a bit, making its effective
bit-depth 18, far exceeding its expected performance. For a higher spatial
resolution SLM (more pixels) these gains are even more impressive; the
Meadowlark SLM with 1920x1152 pixels can be reduced, without too much spatial
resolution loss, to 800x800 pixels. An 8-bit, 800x800 SLM has a minimum step
size of 0.015 zs for 800 nm light if one pixel flips. This gives another order
of magnitude gain, meaning steps of 1 zs could be reliably taken, which is far
beyond what we have the ability to measure.

Experimental demonstrations of the averaging effect were tested with two of
our SLMs. In Fig.5 we show phase scans with step sizes smaller than 1 bit with
both the Meadowlark (a) and Hamamatsu (b) SLMs. The Meadowlark used a step
size of 0.34 bits = 0.0083 rads of the fundamental while the Hamamatsu had
steps of 0.17 bits = 0.0042 rads of the fundamental. Phase control exceeding 1
bit is clearly demonstrated. These step sizes were chosen as they are close to
the limits of what can be measured in these setups due to noise sources such
as vibrations and beam pointing. Our experimental results show that, at least
for linear functions, the required averaging does happen and details of less
than 1 bit can be applied to phase masks with measured optical effects.

Figure 6:  (a) A simulated sine wave phase effect of amplitude 0.51 bits
applied along one axis of a 500x500pixel SLM. In black is the desired phase
that would be applied to each row of the SLM. Red shows what phase would be
applied without this technique. Only the peaks and valleys of the wave are
large enough to be displayed. The purple line shows what phase is displayed if
the linear slopes are subtracted off and the center 16 rows of the mask are
averaged together. The slopes used were $\pi/50$ and $\sqrt{2}/50$ bits per
pixel. There is a significant increase to the precision of phase that can be
applied when pixels are averaged together. (b) How the precision of the
applied phase increases when more pixels are averaged together. Blue shows the
average RMS difference between the desired phase and the averaged phase. Red
shows the equivalent bit-depth increase that would be needed to apple the
phase this precisely.

Although this method was only experimentally tested with flat phase offsets it
can be applied to more complicated phases. Indeed, any phase function can
always be approximated to discrete sections of pixels that are relatively flat
in phase. Depending on the specific effect in question these sections might be
small, only a few pixels, or much larger, with hundreds of pixels depending on
the ideal compromise between spatial and phase resolution. The gain in phase
resolution scales roughly with $N_{p}$, the number of pixels used in the
average. When all sections are combined together the total phase effect should
be more precisely represented on the SLM, though this improvement is difficult
to quantify as it depends heavily on the specific effect being applied. This
enhancement is already helping any system that uses linear slopes for a zeroth
order offset but ensuring that slopes are applied along both SLM axes and that
the slopes depend on distinct irrational numbers is a simple change that will
increase these benefits.

To test this technique with more complicated phase masks we simulated a mask
with a sine wave of amplitude 0.51 bits along one axis of a 500x500 SLM. The
desired phase function is shown in black in Fig. 6 (a). As shown in Fig. 6(a)
in red, an 8-bit representation poorly approximates the true sine wave. When
irrational slopes, of $\pi/50$ and $\sqrt{2}/50$ bits per pixel, are applied
and 16 pixels along the other axis are averaged together the wave is well
approximated, shown in purple. Fig. 6(b) shows, in blue, how the average root
mean squared (RMS) difference between the displayed phase and desired phase
decreases as more pixels are averaged together. Also shown, in red, is the
equivalent increase in bit-depth that would be required to display the mask
that precisely. For full 2-D phases instead of pixels in a line arbitrary 2D
sections of pixels with similar values have to be used.

##  5 Conclusion

We have presented a technique that allows SLMs to overcome their bit-depth
limitations and experimentally tested it with multiple SLMs. The method relies
on using zeroth order offsets yielding irrationally valued slopes and mask
values that are uniformly distributed and with densely packed fractional parts
when averaged locally over a subset of pixels. The fractional parts of any
additional phase, when locally averaged with such slopes, are experimentally
measured to have features of higher precisions than the bit-depth. This was
examined in detail in attosecond interferometric measurements where the
minimum phase delay was improved by a factor of 3, and theoretically predicted
to allow an 8-bit SLM to become effectively 18-bit. With this method two
limiting factors of SLMs are accounted for at once; elimination of the zeroth
order reflection and overcoming bit-depth limits. Increase in phase resolution
comes at the cost of spatial resolution (less effective pixels) but does not
affect the bit rate and the refresh speed. Our approach matches modern SLM
trends of increased number of pixels, faster bit rates but decreased bit depth
and thus can lead to a general approach in the industry.

##  6 Acknowledgements

We thank fruitful discussions with Peter Brunt of AVR Optics. Research
supported by US Department of Energy, Office of Science, Chemical Sciences,
Geosciences, & Biosciences Division grant DE-SC0019098. B. D. was supported by
the Office of Naval Research, Directed Energy Ultra-Short Pulse Laser Division
grant N00014-19-1-2339.

##  7 Disclosures

The authors declare no conflicts of interest.

Data availability.  Data underlying the results presented in this paper are
not publicly available at this time but may be obtained from the authors upon
reasonable request.

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[a]Víctor Enguita-Vileta

# On Discrete Goldstone Bosons

Belén Gavela  Rachel Houtz  Pablo Quílez

###### Abstract

Exact discrete symmetries, if non-linearly realized, can reduce the
ultraviolet sensitivity of a given theory. The scalars stemming from
spontaneous symmetry breaking are massive without breaking the discrete
symmetry, and those masses are protected from divergent quadratic corrections.
This is in contrast to non-linearly realized continuous symmetries. The
symmetry-protected masses and potentials of those discrete Goldstone bosons
offer promising physics avenues, both theoretically and in view of the
blooming experimental search for ALPs. In this text, we develop this
theoretical setup for the specific case of a triplet of $A_{5}$ using
invariant theory, showcasing the substantial improvements and compelling
phenomenological consequences introduced by the invariance under a discrete
symmetry.

##  1 Introduction

This text is a summary of the work presented in [1], where a new class of
light scalar particles is introduced and analyzed in detail. We label them
discrete Goldstone bosons (DGBs).

DGBs are a subset of pseudo Nambu-Goldstone bosons (PNGB) whose scalar
potential preserves a discrete symmetry exactly. The main consequence of this
invariance is a substantially reduced UV sensitivity in DGBs, in relation to
the generic case. More specifically, for a generic PNGB, quantum corrections
quadratic in an UV scale $\Lambda$ may introduce large corrections to their
masses and couplings. Meanwhile, corrections of that kind are completely
absent in the DGB potential. This translates into more stable masses which
contain logarithmic contributions with $\Lambda$ at most.

Such an improved UV (in)sensitivity offers an array of opportunities in model-
building. We highlight the case of composite Higgs models [2, 3, 4] as
solutions to the Hierarchy problem. These models generically lack a mechanism
to fully suppress the $\Lambda^{2}$ corrections to the Higgs potential, often
introducing other fine tuning problems in their attempt to solve the initial
puzzle. Interesting applications may exist also in the context of axions,
ALPs, dark matter and other theories containing light scalar particles.

Precedents of our work can be found in Refs. [5, 6]. In our paper, we consider
new specific models and identify novel phenomenological features of DGBs. The
models we present are proof-of-principle, and we leave more realistic
implementions to be presented in further works.

##  2 Non-linearly realized discrete symmetries

Consider a discrete group $D$ and a scalar field $\Phi$ in one of its
$m$-dimensional irreducible representations, then
$\Phi\equiv\left(\phi_{1},\phi_{2},\ldots,\phi_{m}\right)$. If $D$ is
spontaneously broken, the degrees of freedom in $\Phi$ are tied up by the
quadratic restriction

$\Phi^{T}\Phi=\phi_{1}^{2}+\phi_{2}^{2}+\ldots+\phi_{m}^{2}=f^{2}\,,$ (1)

where $f$ is the scale of spontaneous symmetry breaking (SSB). We call this
relation the _non-linearity constraint_ and argue that, if $D$ is an exact
invariance of the scalar potential $V(\Phi)$, then Eq. (1) is enough to forbid
quadratic corrections with UV scales to all levels in the theory. In this text
we will show how is this the case by analyzing the particular example in which
$D=A_{5}$ and $\Phi$ is a triplet. We will also look at the further
phenomenological consequences granted by $A_{5}$-invariance.

How are Goldstone bosons to appear if $D$ is a discrete symmetry, rather than
a continuous one? We stress the fact that any discrete group $D$ is embedded
in a larger continuous one. In our realizations, the continuous group is an
approximate symmetry of the theory, which can be regarded as mildly broken
into $D$111In [1], we develop several example UV completions leading to such a
setup, where the $D$-preserving contributions which break the continuous
symmetry explicitly have a power-like suppression with some perturbative
coefficient, in a way reminiscent of Clockwork [7].. Therefore the Goldstone
Theorem applies.

##  3 A triplet of $A_{5}$

Among the examples examined in [1] is that of the triplet of $A_{5}$, which
has shown to include a number of compelling traits accessory to the UV
stability already discussed.

### Invariants

As to find the building blocks of the $A_{5}$-preserving potential, we use
elements of finite group invariant theory. A triplet representation has three
elementary invariants called the _primary invariants_ ; they are such that all
the remaining invariants can be written as a function of these three. The
_Molien generating function_ [8, 9], in this case given by

$\mathcal{F}_{A_{5}}(\mathbf{1},\mathbf{3};\lambda)=\frac{1+\lambda^{15}}{\left(1-\lambda^{2}\right)\left(1-\lambda^{6}\right)\left(1-\lambda^{10}\right)}\,,$
(2)

is such that the exponents in its denominator equal the order $n$ of the
primary invariants: in this case $n=2\,,6\,,10$. Since $A_{4}\subset A_{5}$, a
clever way of writing down these invariants is to write them in terms of those
of $A_{4}$. We write

$\displaystyle\mathcal{I}_{2}=\Phi^{T}\Phi=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}\,,$
(3)
$\displaystyle\mathcal{I}_{6}=22\mathcal{J}_{3}^{2}+\mathcal{J}_{2}\mathcal{J}_{4}-2\sqrt{5}\mathcal{J}_{6}\,,$
$\displaystyle\mathcal{I}_{10}=\mathcal{J}_{2}^{2}\mathcal{J}_{4}+38\mathcal{J}_{3}^{2}\mathcal{J}_{4}-\frac{7}{11}\mathcal{J}_{2}^{3}\mathcal{J}_{4}-\frac{128}{11\sqrt{5}}\mathcal{J}_{2}^{2}\mathcal{J}_{6}+\frac{6}{\sqrt{5}}\mathcal{I}_{4}\mathcal{I}_{6}\,,$

where $\mathcal{J}_{2}=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}$,
$\mathcal{J}_{3}=\phi_{1}\phi_{2}\phi_{3}$,
$\mathcal{J}_{4}=\phi_{1}^{4}+\phi_{2}^{4}+\phi_{3}^{4}$ are the primary
invariants of $A_{4}$ and $\mathcal{J}_{6}$ fulfils the _syzygy_
$4\mathcal{J}_{6}^{2}=2\mathcal{J}_{4}^{3}-5\mathcal{J}_{4}^{2}\mathcal{J}_{2}^{2}+4\mathcal{J}_{4}\mathcal{J}_{2}^{4}-36\mathcal{J}_{4}\mathcal{J}_{3}^{2}\mathcal{J}_{2}-\mathcal{J}_{2}^{6}+20\mathcal{J}_{3}^{2}\mathcal{J}_{2}^{3}-108\mathcal{J}_{3}^{4}$.

From the numerator in Eq. (2), one also infers the existence of an invariant
of order $n=15$, $\mathcal{I}_{15}$, which can be written as a non-polynomial
function of the primary ones. All the remaining invariants of $A_{5}$ can be
written as a polynomial in $\mathcal{I}_{2}$, $\mathcal{I}_{6}$,
$\mathcal{I}_{10}$ and $\mathcal{I}_{15}$.

Figure 1: Manifold defined by the $\mathcal{I}_{6}(\Phi)$ and
$\mathcal{I}_{10}(\Phi)$ invariants of $A_{5}$, with $\Phi$ a triplet
fulfilling $\mathcal{I}_{2}(\Phi)=\Phi^{T}\Phi=f^{2}$. A smaller discrete
symmetry, denoted in parenthesis, remains explicitly realized after SSB around
the MaNa extrema, $A$, $B$ and $C$.

### EFT

By Eq. (1), after spontaneous symmetry breaking
$\mathcal{I}_{2}=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=f^{2}$. That is, _the
invariant of order $n=2$ is rendered non-dynamical_. Since $\mathcal{I}_{2}$
is the only invariant of order $n=2$ allowed by the discrete symmetry, $A_{5}$
in this case, there are no contributions to to the potential at that order. In
consequence, and simply by dimensional arguments, $\Lambda^{2}$ _contributions
must also be absent_. In fact, there will be no renormalizable contributions
to this potential, which starts at order $n=6$.

The complete $A_{5}$-preserving potential for a triplet $\Phi$ can be written
in terms of of $\mathcal{I}_{6}$ and $\mathcal{I}_{10}$ only,

$V_{\mathrm{DGB}}=f^{2}\Lambda^{2}\sum_{a,b,c}^{\infty}c_{abc}\left(\frac{\mathcal{I}_{6}}{f^{6}}\right)^{a}\left(\frac{\mathcal{I}_{10}}{f^{10}}\right)^{b}\left(\frac{\mathcal{I}_{15}}{f^{15}}\right)^{c}\,,\quad\Lambda\leq
4\pi f\,,$ (4)

where as stated previously
$\mathcal{I}_{15}=\mathcal{I}_{15}(\mathcal{I}_{6},\mathcal{I}_{10})$.

### Maximally natural extrema

The potential in Eq. (4) contains a wealth of possible extrema. They can be
generically calculated by solving

$\frac{\partial V_{\text{DGB}}}{\partial\phi_{j}}=\sum_{i}\frac{\partial
V_{\text{DGB}}}{\partial\mathcal{I}_{i}}\frac{\partial\mathcal{I}_{i}}{\partial\phi_{j}}=\sum_{j}\frac{\partial
V_{\text{DGB}}}{\partial\mathcal{I}_{i}}J_{ij}=0\,,$ (5)

where $J_{ji}=\partial\mathcal{I}_{i}/\partial\phi_{j}$. Extrema that happen
when $\partial V_{\text{DGB}}/\partial\mathcal{I}_{i}=0$ appear in specific
realizations of the theory and are functions of the $c_{abc}$’s. We are
interested in those extrema which correspond to taking $J_{ji}=0$, and label
them as _maximally natural extrema_ (MaNa extrema), since they only depend on
the underlying symmetry of the theory.

Fig. 2 shows the manifold defined by the invariants $\mathcal{I}_{6}$ and
$\mathcal{I}_{10}$ of $A_{5}$ as they take all their possible values. The MaNa
extrema are located on the 0-dimensional intersections between the different
1-dimensional branches of the boundary of the manifold, where the jacobian
must always vanish. It can be shown that around each of those extrema, a
smaller symmetry remains explicitly realized. In the next subsection, we
briefly review the phenomenological consequences of the $Z_{5}$ invariance of
point $B$.

### Phenomenology

Figure 2: Production of DGBs from SM collisions, for a triplet of non-linearly
realized $A_{5}$. Notice that events with three DGBs are absent.

The dominant contribution in Eq. (3) is simply given by the invariant
$\mathcal{I}_{6}$; expanding it around point $B$ in Fig. 2 after SSB we find

$\mathcal{I}_{6}=\frac{32}{5}f^{4}\left[\frac{f^{2}}{32}+\left(\pi_{1}^{2}+\pi_{2}^{2}\right)-\frac{31}{12f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)^{2}-\frac{1}{4f^{3}}\left(\pi_{1}^{5}-10\,\pi_{1}^{3}\pi_{2}^{2}+5\,\pi_{1}\pi_{2}^{4}\right)\right]\,.$
(6)

The DGBs, $\pi_{1}$ and $\pi_{2}$, appear arranged in combinations,
$\mathcal{J}^{Z_{5}}_{2}=\pi_{1}^{2}+\pi_{2}^{2}$ and
$\mathcal{J}^{Z_{5}}_{5}=\pi_{1}^{5}-10\,\pi_{1}^{3}\pi_{2}^{2}+5\,\pi_{1}\pi_{2}^{4}$,
which are invariant under $Z_{5}$. This leads to distinctive phenomenological
consequences. Firstly, the two DGBs are degenerate, as it can be read out from
the mass term in Eq. (6). More interestingly, $Z_{5}$-invariance also
determines which are the interaction terms in the theory and their relative
weights. This leads to definite ratios among production rates
$\sigma(SM\rightarrow n\pi)$ of different numbers of DGBs from the SM. Under
the assumption that the SM is a singlet of $A_{5}$, we have the ratios

$\frac{\sigma(\mathrm{SM}\rightarrow 2\pi)}{\sigma(\mathrm{SM}\rightarrow
4\pi)}=\frac{216(4\pi)^{4}}{19(31)^{2}}\frac{f^{4}}{E_{\mathrm{CM}}^{4}}\,,\quad\frac{\sigma(\mathrm{SM}\rightarrow
4\pi)}{\sigma(\mathrm{SM}\rightarrow
5\pi)}=\frac{19(31)^{2}(8\pi)^{2}}{(45)^{2}}\frac{f^{2}}{E_{\mathrm{CM}}^{2}}\,,$
(7)

which may serve to disentangle $A_{5}$-symmetric DGBs from different
realizations of DGBs or other BSM scalar particles. Notice as well that, since
the exact $Z_{5}$ forbids terms linear in the DGBs from appearing the
Lagrangian, any process involving these particles must have them in pairs at
least. This is in contrast with e.g. axions or ALPs, and provides yet another
handle to differentiate DGBs from other scalar particles.

##  4 Conclusions

We have summarized the work in [1] on discrete Goldstone bosons. The main
theoretical motivation of this novel class of particles is the forbidding of
quadratic corrections with UV scales, which is granted automatically upon the
preservation of an exact discrete symmetry. Interesting phenomenological
consequences ensue, which are generic for all DGBs and we have presented for
the compelling case of a triplet of $A_{5}$.

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