# Looking forward to photon-coupled long-lived particles II: dark axion portal

Krzysztof Jodłowski<EMAIL_ADDRESS>Particle Theory and Cosmology
Group, Center for Theoretical Physics of the Universe, Institute for Basic
Science (IBS), Daejeon, 34126, Korea

###### Abstract

The dark axion portal is a dimension-5 coupling between an axion-like particle
(ALP), a photon, and a dark photon, which is one of the targets of the
intensity frontier searches looking for $\sim\,$sub-GeV long-lived particles
(LLPs). In this work, we re-examine the limits set by existing detectors such
as CHARM and NuCal, and by future experiments such as FASER2, MATHUSLA, and
SHiP. We extend previous works by i) considering several mass regimes of the
Dark Sector (DS) particles, leading to an extended lifetime regime of the
unstable species, ii) including LLPs production occurring in previously
neglected vector meson decays that actually dominate the LLP yield, and iii)
by implementing secondary LLP production. It takes place by Primakoff-like
upscattering of lighter DS species into LLP on tungsten layers of neutrino
emulsion detector FASER$\nu$2\. This process will allow FASER2 to cover a
significant portion of the $\gamma c\tau\sim 1\,\text{m}$ region of the
parameter space that is otherwise difficult to cover due to the large ($\sim
O(100)\,\text{m}$) distance between the primary LLP production point and the
decay vessel, where LLP decays take place, which is required in typical beam-
dumb experiments for SM background suppression.

##  I Introduction

The dark axion portal (DAP) has recently been proposed [1, 2] as a novel
interaction between an axion-like particle (ALP), a dark photon (DP), and
photon induced by interactions in the Dark Sector (DS). Such a mechanism can
take place, e.g., due to 1-loop processes involving massive dark fermions
charged under global Peccei-Quinn symmetry $U(1)_{\text{PQ}}$ and gauge groups
- $U(1)_{\text{Y}}$ (hypercharge), and $U(1)_{\text{Dark}}$ \- which can be
viewed as a generalization of the KSVZ [3, 4] axion to DS containing new
$U(1)_{\text{Dark}}$ gauge group.

Recent works have shown that the dark axion portal may have interesting
astrophysical and cosmological implications that differ from the well-studied
photophilic ALP. DAP can, e.g., provide a new mechanism for the production of
dark matter (DM) [5, 6], facilitate cosmological relaxation [7, 8], and lead
to axion-photon-dark photon oscillation [9, 10, 11, 12, 13], which affects 21
cm observations, supernovae cooling, and light-shining-through-walls
experiments, among other things. Similarly to an ALP, which is one of the main
benchmarks of the searches for $\sim\,$sub-GeV feebly-interacting BSM
particles [14, 15, 16], the DAP detection prospects have been investigated in
B-factories, fixed target neutrino experiments, reactor experiments, and beam
dumps [17, 18, 19].

In particular, it was shown that the long-lived particle (LLP) displaced decay
signature at CHARM, FASER, MATHUSLA and SHiP is particularly effective in
covering the $\sim O(100\,\text{m})$ region of the parameter space. Since
then, the FASER experiment, which began collecting data in 2022 at the start
of Run 3 of the LHC, has undergone an intensive research and development
phase. As a result, a dedicated neutrino emulsion detector FASER$\nu$ [20, 21]
has been installed in front of the main detector. Although its main purpose is
the detection of collider neutrinos [22], because it is made of tungsten
layers, it can also act as a target for the secondary production of LLPs; see
[23, 24] for its impact within non-minimal scalar, vector, and sterile
neutrino portals.

Another development concerning FASER that motivates our analysis is that, as
shown in [24], FASER2 will be sensitive to semi-visible two-body LLP decays.
The final states are an invisible particle (a neutrino or a DS state) and a
single high-energy photon: $E_{\gamma}>0.1\,\text{TeV}$ for FASER2111In fact,
recent work [25] considered an even lower energy threshold,
$E_{\gamma}>1{\text{-}}10\,\text{GeV}$. We follow the original thresholds from
[24], while results for lower energy thresholds, or other changes, can be
easily generated using the modified version of $\tt FORESEE$., and
$E_{\gamma}>0.1\,\text{TeV}$ for FASER$\nu$2\. This opens up a possibility to
constrain BSM scenarios with such semi-visible LLP decays in FASER2, where the
LLP can be produced in either primary [26, 25, 27] or secondary [24]
production processes.

In this work, we study both of them for DAP, extending the results of [18] in
multiple directions, such as: considering single photon LLP decays at FASER
detectors, taking into account secondary LLP production occurring just in
front of the main decay vessel, which allows to cover the regime of shorter
LLP lifetimes, and electron scattering signatures; we note that in upcoming
work we investigate other BSM scenarios using similar signatures [28, 29].
Moreover, we study both LLP candidates - when it consists of a dark axion or a
dark photon, and our simulation is adapted to the general case.

The paper is organized as follows. In Sec. II we discuss the physical aspects
of DAP that are relevant to the intensity frontier searches. In particular, we
identify the region of parameter space corresponding to the long-lifetime
regime of a dark photon or a dark axion. In Sec. III we provide the specifics
of the LLP production modes and the signatures under consideration, such as:
the displaced LLP decays, secondary LLP production, and scattering of DS
states with electrons. Our main results are discussed in Sec. IV. We show
sensitivity reach for FASER2, MATHUSLA, NuCal, and SHiP in two mass
hierarchies, where either the dark photon or the dark axion act as a LLP. For
both scenarios, we consider several fixed values of the mass ratio between the
two DS species, which correspond to different LLP lifetime regimes. We also
compare our results to the case of photophilic ALP. In Sec. V we summarize our
study.

##  II Dark axion portal

The interaction Lagrangian of the dark axion portal is [1, 2],

$\displaystyle\\!\\!\mathcal{L}$
$\displaystyle\supset\frac{g_{a\gamma\gamma^{\prime}}}{4}aF^{\mu\nu}\tilde{F^{\prime}}_{\mu\nu}\,,$
(1)

where $g_{a\gamma\gamma^{\prime}}$ is a coupling of mass-dimension -1, and
$F_{\mu\nu}$ and $F^{\prime}_{\mu\nu}$ are the EM and $U(1)_{\text{Dark}}$
field strength tensors, respectively.

As described in Sec. I, the dark axion portal leads to an interesting range of
phenomena that can be distinct from the photophilic ALP. In particular, dark
axion portal was proposed [17] as an explanation of the recently rejuvenated
$(g-2)_{\mu}$ anomaly [30, 31, 32, 33]. The region of parameter space relevant
to such a solution was the long-lived dark photon with $\sim\,$GeV mass. In
fact, [17] analyzed the dark photon displaced decays in the past beam dump and
neutrino experiments: LSND [34], MiniBooNE [35, 36], and CHARM [37] (missing
energy searches at BaBar [38, 39] and Belle [40] were also considered, and
were shown to provide the coverage of the high mass range) to exclude such
possibility. On the other hand, an extended dark axion portal, involving also
kinetic mixing with SM hypercharge or muon-philic interactions, has been shown
to be a viable solution [41, 42]. Moreover, such a scenario could be tested in
future lepton fixed target experiments, such as NA64$e$ [43, 44], NA64$\mu$
[45], LDMX [46], and M3 [47]. These considerations further motivate dedicated
sensitivity study of the long-lifetime regime of the DAP at the far-forward
region of the LHC.

In the following, we discuss two benchmarks where, in each case, one of the DS
species is massless, stable particle. Then only the coupling
$g_{a\gamma\gamma^{\prime}}$ and the LLP mass are free parameters of the
model. In Sec. IV we present results for both of these benchmarks, as well as
for several additional scenarios in which the masses of the DS states follow a
fixed ratio.

For both benchmarks, the lifetime of the unstable, and typically long-lived,
particle depends on the width of the two-body decay into a photon and a DS
state. The three-body decays into a pair of charged leptons and a DS state are
also possible, especially for $m\gtrsim 0.1\,\text{GeV}$, but they are phase-
space suppressed. As a result, they will contribute to the total decay width
typically only at the $O(0.01)$ level - see Fig. 1 in [18]; relevant formulas
are given in Appendix B.

Since FASER detectors are $\sim 400-600\,\text{m}$ away from the
$p{\text{-}}p$ collision point of the LHC, the typical LLP decay lengths they
can probe are

$\displaystyle d_{\gamma^{\prime}}\simeq$
$\displaystyle\,100\,\text{m}\times\left(\frac{E}{1000\,\text{GeV}}\right)\left(\frac{0.1\,\text{GeV}}{m_{\gamma^{\prime}}}\right)^{4}\left(\frac{7\times
10^{-5}}{g_{a\gamma\gamma^{\prime}}}\right)^{2},$ (2)

for the massless dark axion, while for the massless dark photon analogous
formula for $d_{a}$ holds for $g_{a\gamma\gamma^{\prime}}=4\times
10^{-5}\,\text{GeV}^{-1}$. We used $d=c\beta\tau\gamma$, where $\gamma=E/m$ is
the boost factor of LLP in the LAB frame, $\beta=\sqrt{1-1/\gamma^{2}}$, and
$\tau=1/\Gamma$. The decay widths for the two-body final states are [1]

${\Gamma_{\gamma^{\prime}\to\gamma
a}=\frac{g_{a\gamma\gamma^{\prime}}^{2}}{96\pi}m_{\gamma^{\prime}}^{3}\left(1-\frac{m^{2}_{a}}{m^{2}_{\gamma^{\prime}}}\right)^{3},}\\\
{\Gamma_{a\to\gamma\gamma^{\prime}}=\frac{g_{a\gamma\gamma^{\prime}}^{2}}{32\pi}m_{a}^{3}\left(1-\frac{m^{2}_{\gamma^{\prime}}}{m^{2}_{a}}\right)^{3}.}$
(3)

We note that when one DS particle is massless, the lifetime of $a$ is smaller
than the lifetime of $\gamma^{\prime}$ by a factor of $3$, coming from the
average over dark photon polarization states. The same factor will occur for
other pairs of processes in which $a$ and $\gamma^{\prime}$ are exchanged
which will influence our results in Sec. IV.

Figure 1: Dark photon production modes as a function of its mass. The vector meson decays, which were not included in previous works, dominate at both FASER2 and SHiP. The same relationship exists for other mass schemes and for the dark axion production yields.  Experiment |  Target for prim./sec. prod. | Energy |  Lumi. or $N_{\mathrm{prot.}}$ |  Transverse size | $x_{\mathrm{min}}$ | $\Delta$ | LLP signature |  LLP signature cuts | Ref.
---|---|---|---|---|---|---|---|---|---
CHARM | Cu/- | 400 GeV | $2.4\times 10^{18}$ |  $3\times 3$ m2222CHARM was placed at a distance of 5 m from the beam axis. | 480 m | 35 m | decay |  $E_{e^{+}e^{-}}>3\ \text{GeV}$: $N_{\mathrm{ev}}=3$ $E_{\gamma}>7.5\ \text{GeV}$: $N_{\mathrm{ev}}=100$ | [17, 48]
NuCal | Fe/- | 69 GeV | $1.7\times 10^{18}$ |  $r=1.3$ m | 23 m | 64 m | decay |  $E_{e^{+}e^{-}}>10\ \text{GeV}$: $N_{\mathrm{ev}}=4.4$ $E_{\gamma}>10\ \text{GeV}$: $N_{\mathrm{ev}}=4.4$ | [49, 48]
SHiP | Mo/- | 400 GeV | $2.4\times 10^{18}$ |  $2.5\times 5.5$ m2 | 52.7 m | 50 m | decay |  $E_{e^{+}e^{-}}>3\ \text{GeV}$: $N_{\mathrm{ev}}=3$ $E_{\gamma}>2\ \text{GeV}$: $N_{\mathrm{ev}}=100$ | [17, 23, 25]
FASER2 | Fe333By primary LLP production at the LHC, we mean the Primakoff process in which photons produced in pp collisions hit the iron hadronic absorber TAN located 140 m further converting into a LLP particle; the same is assumed for other vesions of FASER detector. This production mode has been used for photophilic ALP [50] and massive spin-2 portal [51]./- |  $\sqrt{s}=$ $13\,\text{TeV}$ |  $3000$ fb-1 |  $r=1$ m | 480 m | 5 m | decay |  $E_{e^{+}e^{-}}>100\ \text{GeV}$: $N_{\mathrm{ev}}=3$ $E_{\gamma}>100\ \text{GeV}$: $N_{\mathrm{ev}}=3$ | [50, 24]
FASER$\nu$2 | Fe/W |  $\sqrt{s}=$ $13\,\text{TeV}$ |  $3000$ fb-1 |  $r=0.25$ m | 472 m | 2 m |  decay, sec. prod., $e^{-}$ scat. |  $E_{\gamma}>1000\ \text{GeV}$: $N_{\mathrm{ev}}=3$, $300\,\text{MeV}<E_{e^{-}}<20\,\text{GeV}$:444For FASER$\nu$2 and FLArE we also take into the account the angular cuts - see tables 1 and 2 from [52]. $N_{\text{ev}}=20$ | [23, 24]
FPF FASER2 | Fe/- |  $\sqrt{s}=$ $13\,\text{TeV}$ |  $3000$ fb-1 |  $r=1$ m | 620 m | 25 m | decay |  $E_{e^{+}e^{-}}>100\ \text{GeV}$: $N_{\mathrm{ev}}=3$ $E_{\gamma}>100\ \text{GeV}$: $N_{\mathrm{ev}}=3$ | [50, 53, 24]
FPF FASER$\nu$2 | Fe/W |  $\sqrt{s}=$ $13\,\text{TeV}$ |  $3000$ fb-1 |  $0.4\times 0.4$ m2 | 612 m | 8 m |  decay, sec. prod., $e^{-}$ scat. |  $E_{\gamma}>1000\ \text{GeV}$: $N_{\mathrm{ev}}=3$, $300\,\text{MeV}<E_{e^{-}}<20\,\text{GeV}$: $N_{\text{ev}}=20$ | [23, 53, 24]
FPF FLArE | Fe/Ar |  $\sqrt{s}=$ $13\,\text{TeV}$ |  $3000$ fb-1 |  $1\times 1$ m2 | 600 m | 7 m |  sec. prod., $e^{-}$ scat. |  $30\,\text{MeV}<E_{e^{-}}<1\,\text{GeV}$: $N_{\mathrm{ev}}=20$ | [52, 54, 27, 53]
Table 1:  Technical parameters of the considered detectors sensitive to LLP
decays, secondary LLP production or scattering with electrons. We specify the
technical parameters with the references of each experiment used in our
simulations. LHC-based detectors are separated from experiments using
dedicated proton beams. See Tab. 1 from [51] for a table of experiments
sensitive to a massive spin-2 particle decaying dominantly into a photon pair.

Figure 2:  Sensitivity reach for the dark photon acting as the LLP at the
baseline (left) and the Forward Physics Facility (right) location of FASER2.
The mass ratio, $m_{a}/m_{\gamma^{\prime}}$, is fixed as follows: $0$ (top),
$0.5$ (middle), and $0.9$ (bottom). The contour lines for each experiment
correspond to the number of events, $N_{\mathrm{ev}}$, as indicated in table
1. Lines derived by the missing energy signature at BaBar and Belle were taken
from [17].

Figure 3:  Same as Fig. 2 but for dark axion acting as the LLP. We only show
results for one mass scheme, massless dark photon, as the other results are
analogous to the middle and bottom plots of Fig. 2. The light-gray areas are
excluded by astrophysical and cosmological bounds which were obtained in [13].

##  III LLP signatures

In this section, we describe the LLP signatures we use to constrain the DAP,
followed by details of the beam dump and LHC experiments under consideration.

###  III.1 LLP production

In the ${\sim}1\,\text{MeV}{\text{-}}1\,\text{GeV}$ mass range, dark axion and
dark photon are mainly produced from decays of unstable mesons. Compared to
photophilic ALP, for which the Primakoff conversion of an on-shell photon into
ALP dominates, the result is about an order of magnitude smaller number of
LLPs for DAP.

Moreover, with regard to meson decays, previous works [17, 18] considered only
three-body decays of pseudoscalar mesons. As discusssed in [55, 25] for dark
fermions coupled to the SM via dimension-5 and dimension-6 electromagnetic
form factors, the branching ratios of decays of pseudoscalar and vector mesons
of mass $M$ into such DS states are approximately proportional to $M^{2}$. As
a result, the heaviest vector meson produced in sufficiently large quantities
dominate the pseudoscalar mesons contributions. We found agreement with this
argument for the DAP, as shown in Fig. 1, where we present contributions of
unstable mesons produced at the LHC to the dark photon yield assuming the dark
axion is massless; analogous behavior takes place for other mass scenarios.

We used the $\tt FORESEE$ [56] package to implement our model, in particular
we used Eq. 10 and Eq. 11 describing the branching ratios of vector and
pseudoscalar meson decays, respectively, which are used to obtain the
resulting LLP yield. For the far-forward LHC detectors such as FASER2 [57, 58,
59] and Forward Physics Facility (FPF) [60, 61, 53], which would accommodate
multiple detectors adapted to various searches, e.g., FASER$\nu$2 [52, 61] and
FLArE [52], we used the included spectra generated by $\tt EPOSLHC$ [62] and
$\tt Pythia$ [63]. For beam dump experiments such as CHARM [37], NuCal [64,
65], and SHiP [66, 67], we used $\tt Pythia$ to generate the meson spectra,
and we extended $\tt FORESEE$ to simulate the production and decay of LLPs
taking place in these detectors.

###  III.2 Simulation details

After the production of a LLP, the number of events linked to a LLP signature
being detected inside the detector are [68, 69]

$\displaystyle N=\int\int
dEd\theta\frac{d^{2}N}{dEd\theta}\,p(E,\theta)\,q_{\text{accept.}}(E,\theta),$
(4)

where the first term denotes the spectrum of the LLP with a energy $E$ and
polar angle $\theta$ relative to the beamline; $p(E)$ corresponds to the
probability of the signature taking place inside the detector, while
experimental or simulation-related cuts are described by
$q_{\text{accept.}}(E,\theta,\phi)$.

#### Primary production

Displaced LLP decays resulting from, e.g., proton-target collisions are the
main experimental signature in LLP searches [14, 15, 70]. The experimental
signal consists of high-energy SM particles, typically a pair of photons or
charged leptons, and the probability of these decays occurring within a
detector of length $\Delta$ is

$\displaystyle p(E)=e^{-L/d(E)}-e^{-(L+\Delta)/d(E)},$ (5)

where $d(E)$ represents the LLP decay length in the LAB frame and $L$
corresponds to the distance between the LLP production point and the start of
the detector. It is evident that the majority of events arise from
sufficiently long-lived species, characterized by $d\gtrsim L$, resulting only
in linear suppression with the decay length: $p(E)\simeq\Delta/d$ [71, 15].
However, for short-lived species, the second term in Eq. 5 can be neglected
and $p(E)\simeq e^{-L/d}$. It is therefore clear that the distance $L$ sets
the scale of the LLP decay lengths that can be probed in such a way.

In the DAP, the leading two-body decays deposit energy through a single
photon, while decays into a DS and $e^{+}e^{-}$ are suppressed, see bottom
panels of each plot in Figs. 2 and 3. Despite the additional SM induced
background for the single-photon LLP decay, it was shown [24] that FASER2 will
be sensitive to it with the same cuts on the deposited energy and number of
events as for the two-photon decays; we refer to that work for discussion of
the backgrounds.

#### Secondary production

Secondary production of LLPs can take place by coherent upscattering of a
lighter DS species into the LLP on tungsten layers of neutrino emulsion
detector FASER$\nu$2; see fig. 1 from [23] for a schematic illustration.

We study the displaced decay of the LLP produced in this way, where the
production takes place at FASER$\nu$2, while the decay happens at FASER2. As
the distance between these two detectors is $L\simeq 1\,\text{m}$, this
production mode could allow to cover a part of the $d\sim L\simeq 1\,\text{m}$
region of the parameter space. On the other hand, the cross-section for the
secondary production results in additional $\propto
g_{a\gamma\gamma^{\prime}}^{2}$ dependence in the number of decays. As a
result, we expect secondary production to cover larger values of
$g_{a\gamma\gamma^{\prime}}$ than the ones covered by primary production [23,
24].

The probability of secondary LLP production followed by decay inside FASER2 is
given by convolution of Eq. 5 with upscattering cross-section [23]

$\displaystyle p(E)_{\text{sec.\\!
prod.}}=\frac{1}{L_{\text{int}}}\int_{0}^{\tilde{\Delta}}\left(e^{-(x_{\text{min}}-t)/d}-e^{-(x_{\text{min}}+\Delta-t)/d}\right)\,dt=\frac{d}{m_{T}/(\rho\,\sigma(E))}e^{-(x_{\text{min}}+\Delta)/d}\left(e^{\Delta/d}-1\right)\left(e^{\tilde{\Delta}/d}-1\right),$
(6)

where $L_{\text{int}}=m_{T}/(\rho\,\sigma(E))$ is the interaction length
corresponding to the upscattering of DS species with energy $E$ on nucleus of
mass $m_{T}$ inside the material of density $\rho$ and length
$\tilde{\Delta}$; $\sigma(E)$ is the upscattering cross-section;
$x_{\text{min}}$ is the distance from the beginning of the upscattering
material to the beginning of the detector; and the dummy variable $t$
parameterizes the length of the upscattering material.

The cross-section for the upscattering process can be obtained in the closed
form following the method described for photophilic ALP [72]; also see eq.
B1-B3 from [51], while the derivation of the equation below can be found in
the included Mathematica notebook,

$\sigma_{\gamma^{\prime}N\to
aN}\simeq\frac{\alpha_{\mathrm{EM}}g_{a\gamma\gamma^{\prime}}^{2}Z^{2}}{12}\left(\log\left(\frac{d}{1/a^{2}-t_{\mathrm{max}}}\right)-2\right),$
(7)

where $a=111Z^{-1/3}/m_{e}$, $d=0.164\,\text{GeV}^{2}A^{-2/3}$, $m_{e}$ is the
electron mass, $Z$ ($A$) is the atomic number (weight) of a nucleus, and
$t_{\mathrm{max}}\simeq-(m_{a}^{4}+m_{\gamma^{\prime}}^{4})/(4E_{1}^{2})$.

This formula differs (it is smaller) from the one for photophilic ALP only by
a factor of $2/3$, which results from $2$ ($3$) polarization states for photon
(dark photon).

At FASER$\nu$2, Primakoff production process takes place on tungsten (W), and
the formula describing it has the following form:
$\sigma^{\text{W}}_{\gamma^{\prime}N\to
aN}\simeq\frac{48}{\mathrm{GeV}^{2}}\times\left(\frac{g_{a\gamma\gamma^{\prime}}}{1/\mathrm{GeV}}\right)^{2}$.

#### Electron scattering

FASER$\nu$2 and FLArE detectors will be sensitive to DS states scattering with
electrons, see [52] for an extensive discussion.

The corresponding probability for such scattering events is simply given by

$\displaystyle p(E)_{\text{scat.}}=\frac{\Delta}{L_{\text{int}}},$ (8)

where $\Delta$ is the length of the FASER$\nu$2 or FLArE and $L_{\text{int}}$
denotes the interaction length of the scattering process, which is described
by the following cross-section:

$\sigma_{\gamma^{\prime}e^{-}\to
ae^{-}}\simeq\frac{\alpha_{\mathrm{EM}}g_{a\gamma\gamma^{\prime}}^{2}}{12}\log\left(\frac{E_{R}^{max}}{E_{R}^{min}}\right),$
(9)

where $E_{R}\equiv E_{e^{-}}$ is the electron recoil energy, while the
differential cross-section, $d\sigma/dE_{R}$, needed for angular cuts
indicated in table 1, can be found in the Mathematica notebook.

##  IV Results

We present the sensitivity lines for the LLP signatures within DAP in past and
future experiments listed in table 1. In the upper rows, the characteristics
of beam dump experiments are displayed, whereas the lower rows exhibit the
far-forward LHC detectors. We consider two versions of the FASER2 detector -
an extension of current FASER detector or new one placed within FPF. We
indicated all their relevant properties needed for simulation of LLP
signatures discribed in Sec. III.

Although, as indicated by similar formulas for decay widths and Primakoff
cross-sections for both DAP and photophilic ALP, the main difference between
the two portals is the fact that DAP is composed of two DS species and,
depending on the mass hierarchy between them, each can serve as a LLP.

In Sec. IV.1 we discuss the case when dark photon is the LLP, which for
massless dark axion was studied in [17, 18, 19], while in Sec. IV.2 we
consider dark axion as the LLP.

###  IV.1 Dark photon as the LLP

When $m_{\gamma^{\prime}}>m_{a}$, $\sim\,$sub-GeV dark photon is the LLP, and
its decay width is described by Eq. 2. The signatures described in Sec. III
were simulated in modified version of $\tt FORESEE$, and the results are shown
in Fig. 2.

We consider three mass ratios, $m_{a}/m_{\gamma^{\prime}}$, which are fixed as
follows: $0$ (top), $0.5$ (middle), and $0.9$ (bottom).

In the first case, we checked that when the dark photon is produced only by
the three-body pseudoscalar meson decays, we reproduce the results of [17,
18]. Moreover, for the case of massless dark axion (and also in the opposite
case of massless dark photon), we denote with the light-gray color the areas
that are excluded by astrophysical and cosmological bounds obtained in [13].

The richness of the DAP is indicated by the middle and bottom plots of Fig. 2.
They show that when the masses of the two DS particles are comparable, the LLP
decay width is suppressed and, as a result, its lifetime is longer, resulting
in shifting the significant reach of FASER2 and SHiP towards higher masses.
Note that in this scenario the existing bounds, especially from NuCal, are
relaxed due to the high energy threshold on the single photon, which is more
difficult to meet because of the compressed spectra. On the other hand, FASER2
reach weakens only mildly because of the typical high energy $\sim
O(100)^{\prime}s\,\text{GeV}$ of the produced LLPs.

Another feature of DAP that distinguishes it from the photophilic ALP is that
vector meson decays produce a pair of dark photon-dark axion, both of which
can travel virtually undisturbed from the production point to FASER$\nu$2,
which allows for the secondary LLP production by Primakoff-like
upscattering555It also increases the number of events from electron scattering
by a factor of 2.. As a result, this production mode allows to cover part of
the $d=\gamma c\tau\sim 1\,\text{m}$ region of the parameter space, see dashed
and dash-dotted lines in Figs. 2 and 3. Note that the probability of LLP decay
taking place inside the decay vessel in short-lived regime is exponentially
suppressed, $p(E)\simeq e^{-L/d}$ for $d\ll L$, hence this region of the
parameter space cannot be covered by a detector placed at a significant
distance from the LLP production point.

Lastly, the electron scattering signature allows coverage of the low-mass
regime and is complementary to the decays of the dark photons produced in both
primary and secondary production processes. It should be noted that the
electron scattering limit is typically weaker than in the case of secondary
production, mainly due to the lack of $Z^{2}$ enhancement, cf. Eq. 7 and Eq.
9.

###  IV.2 Dark axion as the LLP

The results for the opposite mass hierarchy are shown in Fig. 3. The formulas
for the LLP production channels are the same as for the case of the dark
photon acting as the LLP, while the LLP lifetime is smaller by a factor of
$3$; see Eq. 3. As a result, the sensitivity lines are shifted towards smaller
masses. Moreover, the Primakoff and electron scattering cross-sections are
also smaller by a factor of $3$, resulting in smaller reach.

We only show one benchmark corresponding to massless dark photon, while other
mass scenarios are analogous to the middle and bottom rows of Fig. 2.

##  V Conclusions

In this paper, we have studied the prospects of detecting the dark axion
portal in the intensity frontier searches adapted to a diverse set of LLP
signatures. The main difference between DAP with negligible kinetic mixing and
the photophilic ALP in the $\sim\,$sub-GeV mass regime is that the Primakoff
conversion of an on-shell photon into an ALP is no longer possible, and the
leading LLP production modes are vector meson decays, yielding approximately
an order of magnitude fewer LLPs.

The second difficulty in probing DAP is that the LLP decays semi-visibly, so
its energy can only be deposited by a single photon666As Figs. 2 and 3 show,
three-body semi-visible LLP decays into a $e^{+}e^{-}$ pair are suppressed by
at least 2 orders of magnitude with respect to the leading two-body decays..
Such an experimental signature is more challenging than the usual two-photon
ALP decay because of, among other things, the additional SM induced
background.

On the other hand, future detectors like FASER2 and SHiP will be able to
effectively probe such LLP decays, resulting in coverage of the parameter
space similar to the photophilic ALP.

Moreover, secondary LLP production taking place just in front of the decay
vessel will allow to cover part of the shorter LLP lifetime regime
corresponding to $d\sim 1\,\text{m}$.

Finally, scatterings of either of the DS species with electrons taking place
inside FASER$\nu$2 or FLArE will allow to probe the low mass regime of the
LLP, $m\lesssim 10\,\text{MeV}$. It is complementary to both the LLP displaced
decays and missing energy searches, which are restricted to different LLP mass
ranges.

###### Acknowledgements.

This work was supported by the Institute for Basic Science under the project
code, IBS-R018-D1.

##  Appendix A Meson decays

In this section, we present expressions for the leading dark axion and dark
photon production modes, see Fig. 1 for a comparison of their contributions.

###  A.1 Vector meson decays

We give the branching ratio of the two-body decays of vector mesons into
$a{\text{-}}\gamma^{\prime}$,
$V(p_{0})\\!\to\\!\gamma^{*}(p_{1}+p_{2})\\!\to\\!a(p_{1})+\gamma^{\prime}(p_{2})$,
which are mediated by an off-shell photon,

$\displaystyle\frac{{\rm BR}_{V\rightarrow a\gamma^{\prime}}}{{\rm
BR}_{V\rightarrow
ee}}=\frac{g_{a\gamma\gamma^{\prime}}^{2}\left((-M^{2}+m_{a}^{2}+m_{\gamma^{\prime}}^{2})^{2}-4m_{a}^{2}m_{\gamma^{\prime}}^{2}\right)^{3/2}}{32\pi\alpha_{\text{EM}}M\sqrt{M^{2}-4m_{e}^{2}}\left(M^{2}+2m_{e}^{2}\right)},$
(10)

where ${\rm BR}_{V\rightarrow e^{+}e^{-}}$ is the branching ratio of the
vector meson with mass $M$ decaying into the $e^{+}e^{-}$ pair [73].

We note that in the $m_{\gamma^{\prime}}\to 0$ limit Eq. 10 reduces to the
result for photophilic ALP, see, e.g., Eq. 11 from [74].

###  A.2 Pseudoscalar and vector meson decays

The subdominant production mode takes place by decays of pseudoscalar mesons
into a photon and DS states mediated by an off-shell photon,
$P(p_{0})\\!\to\\!\gamma(p_{1})+\gamma^{*}(p_{2}+p_{3})\\!\to\\!\gamma(p_{1})+a(p_{2})+\gamma^{\prime}(p_{3})$.

We obtained the same averaged amplitude squared as [17], while below we give
the resulting differential branching ratio in a form convenient for Monte
Carlo simulation:

$\displaystyle\frac{d{\rm BR}_{P\rightarrow\gamma
a\gamma^{\prime}}}{dq^{2}d\cos\theta}={\rm BR}_{P\rightarrow\gamma\gamma}$
$\displaystyle\\!\times\\!\\!\left[\frac{g_{a\gamma\gamma^{\prime}}^{2}}{256\pi^{2}m_{P}^{6}q^{6}}\left(m_{P}^{2}-q^{2}\right)^{3}(\cos(2\theta)+3)\left((m_{\gamma^{\prime}}^{2}+m_{a}^{2}-q^{2})^{2}-4m_{\gamma^{\prime}}^{2}m_{a}^{2}\right)^{3/2}\\!\right],$
(11)

where $m_{P}$ is the pseudoscalar meson mass, $q^{2}\equiv(p_{2}+p_{3})^{2}$,
and $\theta$ is the angle between $a$($p_{2}$) and the off-shell photon in the
meson rest frame; ${\rm BR}_{P\rightarrow\gamma\gamma}$ is the branching ratio
of pseudoscalar meson decaying into two photons taken for PDG [73].

##  Appendix B Three-body decays

Below, we give formulas for the three-body decay widths of a dark photon and a
dark ALP in the $m_{\gamma^{\prime}}\gg m_{a},m_{l}$ and $m_{a}\gg
m_{\gamma^{\prime}},m_{l}$ limits, respectively; general form of the
differential decay width can be found in the Mathematica notebook included in
.

$\Gamma_{\gamma^{\prime}\to
l^{+}l^{-}a}=\frac{\alpha_{\mathrm{EM}}g_{a\gamma\gamma^{\prime}}^{2}}{576\pi^{2}m_{\gamma}^{3}}\left(32m_{l}^{6}\coth^{-1}\left(\frac{m_{\gamma^{\prime}}}{\sqrt{m_{\gamma^{\prime}}^{2}-4m_{l}^{2}}}\right)+m_{\gamma^{\prime}}\left(\sqrt{m_{\gamma^{\prime}}^{2}-4m_{l}^{2}}\left(26m_{\gamma^{\prime}}^{2}m_{l}^{2}-7m_{\gamma^{\prime}}^{4}+8m_{l}^{4}\right)-4m_{\gamma^{\prime}}^{5}\log\left(\frac{2m_{l}}{\sqrt{m_{\gamma^{\prime}}^{2}-4m_{l}^{2}}+m_{\gamma^{\prime}}}\right)+12m_{\gamma^{\prime}}m_{l}^{4}\log\left(\frac{16m_{l}^{4}\left(m_{\gamma^{\prime}}-\sqrt{m_{\gamma^{\prime}}^{2}-4m_{l}^{2}}\right)}{\left(\sqrt{m_{\gamma^{\prime}}^{2}-4m_{l}^{2}}+m_{\gamma^{\prime}}\right)^{5}}\right)\right)\right),$
(12)

$\Gamma_{a\to
l^{+}l^{-}\gamma^{\prime}}=\frac{\alpha_{\mathrm{EM}}g_{a\gamma\gamma^{\prime}}^{2}}{192\pi^{2}m_{a}^{3}}\left(32m_{l}^{6}\coth^{-1}\left(\frac{m_{a}}{\sqrt{m_{a}^{2}-4m_{l}^{2}}}\right)+m_{a}\left(\sqrt{m_{a}^{2}-4m_{l}^{2}}\left(26m_{a}^{2}m_{l}^{2}-7m_{a}^{4}+8m_{l}^{4}\right)-4m_{a}^{5}\log\left(\frac{2m_{l}}{\sqrt{m_{a}^{2}-4m_{l}^{2}}+m_{a}}\right)+12m_{a}m_{l}^{4}\log\left(\frac{16m_{l}^{4}\left(m_{a}-\sqrt{m_{a}^{2}-4m_{l}^{2}}\right)}{\left(\sqrt{m_{a}^{2}-4m_{l}^{2}}+m_{a}\right)^{5}}\right)\right)\right).$
(13)

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# Orthogonal Determinants of $\operatorname{SL}_{3}$(q) and
$\operatorname{SU}_{3}$(q)

Linda<EMAIL_ADDRESS>and Gabriele<EMAIL_ADDRESS>aachen.de

(Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Germany)

###### Abstract

We give a full list of the orthogonal determinants of the even degree
indicator ’+’ ordinary irreducible characters of $\mathrm{SL}_{3}(q)$ and
$\mathrm{SU}_{3}(q)$.
Keywords: Orthogonal representations, invariant quadratic forms, generic
orthogonal character table, finite groups of Lie type. MSC: 20C15, 11E12.

##  1 Introduction

Let $G$ be a finite group and $\rho:G\to\mathrm{GL}_{n}(\mathbb{C})$ be a
representation. We call $\rho$ an orthogonal representation, if there is a
symmetric, non-degenerate, $\rho(G)$-invariant bilinear form $\beta$ on
${\mathbb{C}}^{n}$. It is well-known that being an irreducible orthogonal
representation is equivalent to the associated character having Frobenius-
Schur indicator ’+’, i.e. $\rho$ being equivalent to a real representation. A
character is called orthogonal if it is the character afforded by an
orthogonal representation. The character $\chi$ is orthogonal, if and only if
it is of the form

$\chi=\sum_{i=1}^{r}a_{i}\chi^{(+)}_{i}+2\sum_{j=1}^{s}b_{i}\chi^{(-)}_{j}+\sum_{k=1}^{t}c_{i}(\chi^{(0)}_{k}+\overline{\chi_{k}}^{(0)}),$
(1)

where $\chi_{i}^{(+)}$ (resp. $\chi^{(-)}_{j}$, resp. $\chi^{(0)}_{k}$) are
irreducible characters of $G$ with Frobenius-Schur indicator ’+’ (resp. ’-’,
resp. ’0’), and $a_{i},b_{j},c_{k}$ are non-negative integers.

Let $\chi$ be an orthogonal character as in equation (1) and let
$K=\mathbb{Q}(\chi)$ be the character field of $\chi$. The main result of [8]
shows that if the degree of all $\chi_{i}^{+}$ is even then there is a unique
element $\det(\chi)\coloneqq d\in K^{\times}/(K^{\times})^{2}$, called the
orthogonal determinant of $\chi$, such that for all representations
$\rho:G\to\mathrm{GL}_{n}(L)$ over a field extension $L\supseteq K$ affording
$\chi$ all non-degenerate, $\rho(G)$-invariant, symmetric bilinear forms
$\beta$ on $L^{n}$ have the same determinant

$\det(\beta)=d\cdot(L^{\times})^{2}\in L^{\times}/(L^{\times})^{2}.$

We call such a character orthogonally stable.

The orthogonal determinant of $2\chi_{j}^{(-)}$ is always $1$ (see [13]) and
for characters of the form $\chi_{k}^{(0)}+\overline{\chi_{k}}^{(0)}$ the
orthogonal determinant can be obtained from the character values as given in
Lemma 2.4. So it remains to deal with the indicator ’+’ part. Put

$\mathrm{Irr}^{+}(G)=\\{\chi\in\mathrm{Irr}(G)\mid\chi\ \text{is an orthogonal
character of even degree}\\}.$

In a long term project theoretical and computational methods are used to
calculate the orthogonal determinants of the small finite simple groups (see
[2] for a survey). The goal of this paper is to determine the orthogonal
determinants of the characters in $\mathrm{Irr}^{+}(G)$ for the two infinite
series of finite groups of Lie type, $G=\mathrm{SL}_{3}(q)$ and
$G=\mathrm{SU}_{3}(q)$, for all prime powers $q$. For $\mathrm{SL}_{2}(q)$ the
orthogonal character table is already computed in [1]. An important subgroup
to analyse the structure and the representations of a finite group $G$ of Lie
type is its standard Borel subgroup $B$. A character
$\chi\in\operatorname{Irr}^{+}(G)$ is called Borel stable, if the restriction
of $\chi$ to $B$ is orthogonally stable. The structure of $B$ as a semidirect
product of a $p$-group and a split torus $T$ allows us to determine the
orthogonal determinants of all Borel stable characters of $G$ (see Remark
4.2). For the groups $G=\mathrm{SL}_{3}(q)$ and $G=\mathrm{SU}_{3}(q)$ it
turns out that those $\chi\in\operatorname{Irr}^{+}(G)$ that are not Borel
stable occur nicely in a permutation character. For such characters Lemma 2.5
can be used to determine their orthogonal determinants.

This paper is a contribution to Project-ID 286237555 – TRR 195 – by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

##  2 Methods

This section collects some basic results about orthogonal determinants.

######  Definition 2.1.

Let $G$ be a finite group, $H\subseteq G$ a subgroup, and let $\chi$ be an
orthogonal character of $G$. Then $\chi$ is called $H$-stable if the
restriction $\mathrm{Res}^{G}_{H}(\chi)$ of $\chi$ to $H$ is an orthogonally
stable character of $H$.

######  Lemma 2.2.

(see [9, Proposition 5.17 and Remark 5.21] for a more general statement of
(ii))

  1. (i)

If $\chi$ is $H$-stable, then $\chi$ is orthogonally stable and

$\det(\chi)=\det(\mathrm{Res}^{G}_{H}(\chi))\cdot(\mathbb{Q}(\chi)^{\times})^{2}.$

  2. (ii)

If $\chi=\sum_{i=1}^{k}\chi_{i}$ is the sum of orthogonally stable characters
$\chi_{i}$ then $\chi$ is orthogonally stable. Moreover if
$\mathbb{Q}(\chi_{i})\subseteq\mathbb{Q}(\chi)$ for all $i$ then

$\det(\chi)=\prod_{i=1}^{k}\det(\chi_{i})\cdot(\mathbb{Q}(\chi)^{\times})^{2}.$

The paper [7] gives an easy formula for the orthogonal determinant of
orthogonally stable characters of $p$-groups. We only need the following
special case:

######  Lemma 2.3.

(see [7, Corollary 4.4]) Let $p$ be an odd prime and let $\chi$ be an
orthogonally stable rational character of a finite $p$-group. Then $p-1$
divides $\chi(1)$ and
$\det(\chi)=p^{\chi(1)/(p-1)}\cdot(\mathbb{Q}^{\times})^{2}$.

######  Lemma 2.4.

(see [9, Proposition 3.12]) Let $\psi=\chi+\overline{\chi}$ for some indicator
’0’ irreducible character $\chi$. Let $K=\mathbb{Q}(\psi)$,
$L=\mathbb{Q}(\chi)$. Then $K$ is the maximal real subfield of the complex
field $L$. So there is a totally positive $\delta\in K$ such that
$L=K[\sqrt{-\delta}]$. Then

$\det(\psi)=\delta^{\chi(1)}\cdot(K^{\times})^{2}.$

For the cyclotomic fields $L=\mathbb{Q}(\mu_{m}^{j})$, where
$\mu_{m}\coloneqq\exp\left(\frac{2\pi i}{m}\right)\in\mathbb{C}$, we get

$K=\mathbb{Q}(\vartheta_{m}^{(j)})\mbox{ with
}\vartheta_{m}^{(j)}\coloneqq\mu_{m}^{j}+\mu_{m}^{-j}\in\mathbb{R}$

and can choose

$\delta=2-\vartheta^{(2j)}_{m}=-(\mu^{j}_{m}-\mu^{-j}_{m})^{2}.$

######  Lemma 2.5.

Let $G$ be a finite group acting on a finite set $M$. Let $V$ be the
permutation representation over $\mathbb{Q}$. Define the $G$-invariant
bilinear form $\beta:V\times V\to\mathbb{Q}$ by choosing $M$ to be an
orthonormal basis. Then $V_{1}=\langle\sum_{m\in M}m\rangle$ and
$V_{1}^{\bot}$ are $G$-invariant subspaces and

$\det(\beta_{|V_{1}^{\bot}})=|M|\cdot(\mathbb{Q}^{\times})^{2}.$

###### Proof.

It is clear that
$\det(\beta)=\det(\beta_{|V_{1}})\cdot\det(\beta_{|V_{1}^{\bot}})=1\cdot(\mathbb{Q}^{\times})^{2}$
and

$\beta\left(\sum_{m\in M}m,\sum_{m\in M}m\right)=|M|,$

from which the result follows. ∎

##  3 The Orthogonal Characters of $\operatorname{SL}_{3}(q)$ and
$\operatorname{SU}_{3}(q)$

Let $p$ be a prime and let $q$ be a power of $p$. The group
$\mathrm{SL}_{3}(q)$ is

$\mathrm{SL}_{3}(q)=\\{A\in\mathbb{F}_{q}^{3\times 3}|\det(A)=1\\}.$

Let $\mathbb{F}_{q^{2}}\supseteq\mathbb{F}_{q}$ be the field extension of
degree $2$ and let $F:\mathbb{F}_{q^{2}}\to\mathbb{F}_{q^{2}}$, $x\mapsto
x^{q}$ be the Frobenius automorphism. For a matrix
$A\in\mathbb{F}_{q^{2}}^{n\times m}$ we define $F(A)$ to be the matrix where
we apply $F$ component-wise. Let

$\Omega=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix}$

and define the Hermitian form
$H:\mathbb{F}_{q^{2}}^{3}\times\mathbb{F}_{q^{2}}^{3}\to\mathbb{F}_{q^{2}}$,
$H(v,w)=F(v)^{tr}\cdot\Omega\cdot w$. Then

$\displaystyle\mathrm{SU}_{3}(q)=$
$\displaystyle\\{A\in\mathrm{SL}_{3}(q^{2})|H(A\cdot v,A\cdot w)=H(v,w)\
\text{for all}\ v,w\in\mathbb{F}_{q^{2}}\\}=$
$\displaystyle\\{A\in\mathrm{SL}_{3}(q^{2})|F(A)^{tr}\cdot\Omega\cdot
A=\Omega\\}.$

The letter $G$ will always denote one of $\operatorname{SL}_{3}(q)$ or
$\operatorname{SU}_{3}(q)$. The full character table of $G$ was first
calculated in [11] in 1973 by Simpson and Frame. By ’Ennola duality’ (see [4]
for the statement and [6] for a proof),

$"\mathrm{SU}_{3}(q)=\mathrm{SL}_{3}(-q)",$

the irreducible characters of $\mathrm{SU}_{3}(q)$ can be obtained from the
ones of $\mathrm{SL}_{3}(q)$ by formally replacing every instance of $q$ by
$-q$, so that there is a single generic character table giving the character
table for both groups introducing an additional parameter $\varepsilon=+1$ for
$G=\mathrm{SL}_{3}(q)$ and $\varepsilon=-1$ for $G=\mathrm{SU}_{3}(q)$.

In this notation the center of $G$ is the group of scalar matrices in $G$ and
hence of order $\gcd(q-\varepsilon,3)$. In particular the set
$\operatorname{Irr}^{+}(G)$ is the set of irreducible orthogonal characters of
even degree of the group

$\displaystyle\mathrm{PSL}_{3}(q)=\operatorname{SL}_{3}(q)/Z(\operatorname{SL}_{3}(q))\mbox{
and
}\mathrm{PSU}_{3}(q)=\operatorname{SU}_{3}(q)/Z(\operatorname{SU}_{3}(q)).$

It is well known that, with the exception of $\mathrm{PSU}_{3}(2)$, the groups
$\mathrm{PSL}_{3}(q)$ and $\mathrm{PSU}_{3}(q)$ are simple groups for all $q$.
The irreducible characters of $\operatorname{PSU}_{3}(q)$ and
$\operatorname{PSL}_{3}(q)$ are the irreducible characters of $G$ that are
constant on the center.

Gow [5] showed that all the characters of $\mathrm{PSL}_{3}(q)$ and
$\mathrm{PSU}_{3}(q)$ have Schur index 1, except the unique character of
degree $q^{2}-q$ of $\mathrm{PSU}_{3}(q)$, which has Schur index 2 and
Frobenius-Schur indicator ’-’. Additionally, the results in [10] allow us to
obtain the character fields from some combinatorial description. For
cyclotomic numbers we use the notation from Lemma 2.4 and for the naming
convention of the irreducible characters we follow [11, Table 2]. Then the set
$\mathrm{Irr}^{+}(G)$ is given as follows:

######  Theorem 3.1.

The following table includes all $\chi\in\mathrm{Irr}^{+}(G)$, their degrees
$\chi(1)$ and character fields $\mathbb{Q}(\chi)$:

$\chi$ | $u$ | $\chi(1)$ | $\mathbb{Q}(\chi)$
---|---|---|---
$\chi_{qs}$ | — | $q(q+\varepsilon)$ | $\mathbb{Q}$
$\chi_{q^{3}}$ | — | $q^{3}$ | $\mathbb{Q}$
$\chi_{st^{\prime}}^{(u)}$ | $0\leq u\leq 2$ | $1/3(q+\varepsilon)(q^{2}+\varepsilon q+1)$ | $\mathbb{Q}$
$\chi_{st}^{(u,-u,0)}$ |  |  $1\leq u<q-\varepsilon$,
---
$u\notin\\{(q-\varepsilon)/3,(q-\varepsilon)/2,2(q-\varepsilon)/3\\}$
$(q+\varepsilon)(q^{2}+\varepsilon q+1)$ | $\mathbb{Q}(\vartheta_{q-\varepsilon}^{(u)})$
$\chi_{rt}^{((q-\varepsilon)u)}$ | $1\leq u<q+\varepsilon$ | $(q-\varepsilon)(q^{2}+\varepsilon q+1)$ | $\mathbb{Q}(\vartheta_{q+\varepsilon}^{(u)})$

  * •

For $q$ odd and $G=\operatorname{SL}_{3}(q)$,
$\mathrm{Irr}^{+}(G)=\\{\chi_{qs},\chi_{st^{\prime}}^{(u)},\chi_{st}^{(u,-u,0)},\chi_{rt}^{((q-\varepsilon)u)}\\}$.

  * •

For $q$ odd and $G=\operatorname{SU}_{3}(q)$,
$\mathrm{Irr}^{+}(G)=\\{\chi_{st^{\prime}}^{(u)},\chi_{st}^{(u,-u,0)},\chi_{rt}^{((q-\varepsilon)u)}\\}$.

  * •

For $q$ even and $G=\operatorname{SL}_{3}(q)$,
$\mathrm{Irr}^{+}(G)=\\{\chi_{qs},\chi_{q^{3}}\\}$.

  * •

For $q$ even and $G=\operatorname{SU}_{3}(q)$,
$\mathrm{Irr}^{+}(G)=\\{\chi_{q^{3}}\\}$.

Note that the characters $\chi_{st^{\prime}}^{(u)}$ only exist for
$3|q-\varepsilon$.

##  4 Results

Let $G=\mathrm{SL}_{3}(q)$ or $G=\mathrm{SU}_{3}(q)$. Let

$B\coloneqq\left\\{\begin{pmatrix}d&a&b\\\ 0&e&c\\\ 0&0&f\end{pmatrix}\in
G\right\\}\mbox{ and }U\coloneqq\left\\{\begin{pmatrix}1&a&b\\\ 0&1&c\\\
0&0&1\end{pmatrix}\in G\right\\}.$

Then $U$ is the unipotent radical of $B$ and a Sylow $p$-subgroup of $G$, and
$B=N_{G}(U)=U\rtimes T$ is a (standard) Borel subgroup, where
$T:=\\{\mathrm{diag}(d,e,f)\in G\\}$ is a maximal torus. Denote by
$W=N_{G}(T)/T$ the Weyl group of $G$.

We need an explicit notation for $\operatorname{Irr}(T)$:
For $G=\operatorname{SL}_{3}(q)$ we fix a generator $t$ of
$\mathbb{F}_{q}^{\times}$. Then the torus

$T=\\{t_{a,b}:=\mathrm{diag}(t^{a},t^{-a-b},t^{b})\mid
a,b\in\\{0,\ldots,q-2\\}\\}$

is isomorphic to $\mathbb{F}_{q}^{\times}\times\mathbb{F}_{q}^{\times}$ and

$\operatorname{Irr}(T)=\\{\alpha_{1}^{u_{1}}\alpha_{2}^{u_{2}}:t_{a,b}\mapsto\mu_{q-1}^{au_{1}+bu_{2}}\mid
u_{1},u_{2}\in\\{0,\ldots,q-2\\}\\}.$ (2)

For $G=\operatorname{SU}_{3}(q)$ the torus is isomorphic to
$\mathbb{F}_{q^{2}}^{\times}=:\langle\tau\rangle$. So

$T=\\{\tau_{a}:=\mathrm{diag}(\tau^{a},\tau^{(q-1)a},\tau^{-qa})\mid
a\in\\{0,\ldots,q^{2}-2\\}\\}$

and

$\operatorname{Irr}(T)=\\{\alpha^{u}:\tau_{a}\mapsto\mu_{q^{2}-1}^{au}\mid
u\in\\{0,\ldots,q^{2}-2\\}\\}.$ (3)

To unify notation we put

$\theta^{(u)}:=\left\\{\begin{array}[]{ll}\alpha_{1}^{u}\alpha_{2}^{-u}&G=\operatorname{SL}_{3}(q)\\\
\alpha^{u}&G=\operatorname{SU}_{3}(q).\end{array}\right.$ (4)

######  Remark 4.1.

By Lemma 2.2 (i) the orthogonal determinant of a $B$-stable character $\chi$
of $G$ is determined by the restriction of $\chi$ to $B$. Decompose this
restriction as

$\mathrm{Res}^{G}_{B}(\chi)=\chi_{T}+\chi_{U}$ (5)

where $\chi_{T}$ is the character of $T$ on the $U$-fixed space, also known as
the Harish-Chandra restriction of $\chi$. In particular its degree is
$\chi_{T}(1)=\langle\mathrm{Res}^{G}_{U}(\chi),\mathbf{1}_{U}\rangle.$

Note that for odd primes $p$ the character $\chi_{U}$ of $B$ is $U$-stable. As
$p$ does not divide the discriminant of $\mathbb{Q}(\chi)$ for all
$\chi\in\mathrm{Irr}^{+}(G)$ we have $\mathbb{Q}({\chi_{U}})=\mathbb{Q}$ so
Lemma 2.3 gives the determinant of $\chi_{U}$:

######  Remark 4.2.

Let $q$ be odd. If $\chi_{T}$ is orthogonally stable then

$\det(\chi)=\det(\chi_{T})p^{\chi_{U}(1)/(p-1)}$

Note that $T$ is abelian and so $\chi_{T}$ is a sum of linear characters. If
these characters are complex (i.e. of indicator ’0’) then $\chi_{T}$ is
orthogonally stable and its determinant can be computed from Lemma 2.4. In
fact the irreducible constituents of $\chi_{T}$ can be obtained from the
action of the Weyl group $W$ on $\mathrm{Irr}(T)$. It is well-known that

$W\cong\left\\{\begin{array}[]{ll}S_{3}&\mbox{ for }G=\mathrm{SL}_{3}(q)\\\
C_{2}&\mbox{ for }G=\mathrm{SU}_{3}(q)\end{array}\right.$

Let $\theta\in\mathrm{Irr}(T)$. Then $\theta$ can also be considered as a
character of $B$. A character $\chi\in\mathrm{Irr}(G)$ is said to be in the
principal series if $\chi$ appears in $\mathrm{Ind}^{G}_{B}(\theta)$ for some
$\theta\in\mathrm{Irr}(T)$.

We will need a special case of the well-known Mackey formula for Harish-
Chandra induction and restriction:

######  Lemma 4.3.

(see [3, Theorem 5.2.1])

$(\mathrm{Ind}^{G}_{B}(\theta))_{T}=\sum_{w\in W}w\cdot\theta.$

######  Corollary 4.4.

Let $\chi\in\operatorname{Irr}(G)$. Then $0\leq\chi_{T}(1)\leq|W|$, with
$\chi_{T}(1)=0$ if and only if $\chi$ is not in the principal series, and
$\chi_{T}(1)=|W|$ if and only if $\chi=\mathrm{Ind}^{G}_{B}(\theta)$ for some
$\theta\in\operatorname{Irr}(T)$.

Explicit calculations with the character table [11, Table 2] now give rise to
the following propositions:

######  Proposition 4.5.

Let $G=\mathrm{SL}_{3}(q)$. The only characters in $\mathrm{Irr}^{+}(G)$ which
are not in the principal series are $\chi_{rt}^{((q-1)u)}$ for $q$ odd.

  * (a)

$\theta^{(0)}=\mathbf{1}_{T}$ is the trivial character and

$\mathrm{Ind}^{G}_{B}(\theta^{(0)})=\mathbf{1}_{G}+2\chi_{qs}+\chi_{q^{3}}.$

  * (b)

For $1\leq u<q-1$, $u\notin\\{(q-1)/3,(q-1)/2,2(q-1)/3\\}$, we have that

$\mathrm{Ind}^{G}_{B}(\theta^{(u)})=\chi_{st}^{(u,-u,0)}$

and $(\chi_{st}^{(u,-u,0)})_{T}(1)=6$, i.e. the $U$-fixed space in
$\chi_{st}^{(u,-u,0)}$ has dimension 6.

  * (c)

For $j\in\\{(q-1)/3,2(q-1)/3\\}$, we have that

$\mathrm{Ind}^{G}_{B}(\theta^{(j)})=\sum_{u=0}^{2}\chi_{st^{\prime}}^{(u)}$

where $(\chi_{st^{\prime}}^{(u)})_{T}(1)=2$ for $u=0,1,2$.

######  Proposition 4.6.

Let $G=\mathrm{SU}_{3}(q)$. The only characters in $\mathrm{Irr}^{+}(G)$ which
are not in the principal series are $\chi_{st}^{(u,-u,0)}$ and
$\chi_{st^{\prime}}^{(u)}$ for $q$ odd.

  * (a)

$\theta^{(0)}=\mathbf{1}_{T}$ is the trivial character and

$\mathrm{Ind}^{G}_{B}(\theta^{(0)})=\mathbf{1}_{G}+\chi_{q^{3}}.$

  * (b)

For $1\leq u<q+1$, we have that

$\mathrm{Ind}^{G}_{B}(\theta^{((q+1)u)})=\chi_{rt}^{((q+1)u)}$

and $(\chi_{rt}^{((q+1)u)})_{T}(1)=2$.

###### Proof.

We will handle both $G=\mathrm{SL}_{3}(q)$ and $G=\mathrm{SU}_{3}(q)$
simultaneously. Let $d=\gcd(q-\varepsilon,3)$. There are $2+d$ conjugacy
classes of $G$ which have a non-empty intersection with $U$: $C_{1}$, $C_{2}$
and $C_{3}^{(l)}$, $0\leq l\leq d-1$, which can be characterised by
$\mathrm{rank}(g_{i}-I_{3})=i-1$ for $g_{i}\in C_{i},1\leq i\leq 3$. We
further calculate that $|C_{1}\cap U|=1$, and

$\displaystyle|C_{2}\cap U|=\left\\{\begin{array}[]{ll}2q^{2}-q-1&\mbox{ for
}G=\mathrm{SL}_{3}(q)\\\ q-1&\mbox{ for
}G=\mathrm{SU}_{3}(q)\end{array}\right.$ $\displaystyle|C_{3}^{(l)}\cap
U|=\left\\{\begin{array}[]{ll}1/d(q^{3}-2q^{2}+q)&\mbox{ for
}G=\mathrm{SL}_{3}(q)\\\ 1/d(q^{3}-q)&\mbox{ for
}G=\mathrm{SU}_{3}(q).\end{array}\right.$

We will, as an example, calculate
$(\chi_{st}^{(u,-u,0)})_{T}(1)=\langle\mathrm{Res}^{G}_{U}(\chi_{st}^{(u,-u,0)}),\mathbf{1}_{U}\rangle$.
For $\operatorname{SL}_{3}(q)$, we see that

$(\chi_{st}^{(u,-u,0)})_{T}(1)=\frac{1}{q^{3}}\left((q+1)(q^{2}+q+1)+(2q^{2}-q-1)(2q+1)+(q^{3}-2q^{2}+q)\right)=6,$

whereas for $\operatorname{SU}_{3}(q)$, the calculation becomes

$(\chi_{st}^{(u,-u,0)})_{T}(1)=\frac{1}{q^{3}}\left((q-1)(q^{2}-q+1)+(q-1)(2q-1)-(q^{3}-q)\right)=0.$

The rest of the propositions is handled analogously. ∎

We are now ready to give the main result.

######  Theorem 4.7.

  * (i)

Let $q$ be odd.

$\det(\chi)$ for $G=\mathrm{SL}_{3}(q)$ | $\chi$ |  $\det(\chi)$ for $G=\mathrm{SU}_{3}(q)$
---|---|---
$3q\cdot(\mathbb{Q}^{\times})^{2}$ | $\chi_{st^{\prime}}^{(u)}$ | $q\cdot(\mathbb{Q}^{\times})^{2}$
$q(2-\vartheta_{q-1}^{(2u)})\cdot(\mathbb{Q}(\vartheta_{q-1}^{(u)})^{\times})^{2}$ | $\chi_{st}^{(u,-u,0)}$ | $q\cdot(\mathbb{Q}(\vartheta_{q+1}^{(u)})^{\times})^{2}$
$q\cdot(\mathbb{Q}(\vartheta_{q+1}^{(u)})^{\times})^{2}$ | $\chi_{rt}^{((q-\varepsilon)u)}$ | $q(2-\vartheta_{q-1}^{(2u)})\cdot(\mathbb{Q}(\vartheta_{q-1}^{(u)})^{\times})^{2}$

Additionally, for $G=\mathrm{SL}_{3}(q)$,
$\det(\chi_{(qs)})=(q^{2}+q+1)\cdot(\mathbb{Q}^{\times})^{2}$.

  * (ii)

Let $q$ be even.

$\det(\chi)$ for $G=\mathrm{SL}_{3}(q)$ | $\chi$ |  $\det(\chi)$ for $G=\mathrm{SU}_{3}(q)$
---|---|---
$(q+1)(q^{2}+q+1)\cdot(\mathbb{Q}^{\times})^{2}$ | $\chi_{q^{3}}$ | $(q^{3}+1)\cdot(\mathbb{Q}^{\times})^{2}$

Additionally, for $G=\mathrm{SL}_{3}(q)$,
$\det(\chi_{(qs)})=(q^{2}+q+1)\cdot(\mathbb{Q}^{\times})^{2}$.

###### Proof.

We first deal with the case that $q$ is odd: Then, for all
$\chi\in\mathrm{Irr}^{+}(G)\backslash\\{\chi_{qs}\\}$, the value
$\chi_{U}(1)/(p-1)$ is even if and only if $q$ is an even power of $p$. So
Lemma 2.3 gives us that $\det(\chi_{U})=q\cdot(\mathbb{Q}^{\times})^{2}$.

For the characters $\chi_{st}^{(u,-u,0)}$ of $\mathrm{SL}_{3}(q)$ Lemma 4.3
and Proposition 4.5 give that

$(\chi_{st}^{(u,-u,0)})_{T}=\theta_{1}+\overline{\theta}_{1}+\theta_{2}+\overline{\theta}_{2}+\theta_{3}+\overline{\theta}_{3}$

is the sum of $6$ one-dimensional characters of $T$, where
$\theta_{1}=\theta^{(u)}$, $\theta_{2}=\alpha_{1}^{2u}\alpha_{2}^{u}$, and
$\theta_{3}=\alpha_{1}^{u}\alpha_{2}^{2u}$. By Lemma 2.2 and Lemma 2.4

$\det((\chi_{st}^{(u,-u,0)})_{T})=(2-\vartheta_{q-1}^{(2u)})\cdot(\mathbb{Q}(\vartheta_{q-1}^{(u)})^{\times})^{2}$

and

$\displaystyle\det(\chi_{st}^{(u,-u,0)})=$
$\displaystyle\det((\chi_{st}^{(u,-u,0)})_{T})\det((\chi_{st}^{(u,-u,0)})_{U})\cdot(\mathbb{Q}(\vartheta_{q-1}^{(u)})^{\times})^{2}=$
$\displaystyle
q(2-\vartheta_{q-1}^{(2u)})\cdot(\mathbb{Q}(\vartheta_{q-1}^{(u)})^{\times})^{2}.$

The results for $\chi_{rt}^{((q-\varepsilon)u)}$ for $\mathrm{SU}_{3}(q)$ and
$\chi_{st^{\prime}}^{(u)}$ follow similarly. Note here that

$2-\vartheta_{q-1}^{(2(q-1)/3)}=2-(-1)=3.$

The character $\chi_{qs}$ for $G=\mathrm{SL}_{3}(q)$ ($q$ even or odd) occurs
in the permutation character $\psi:=1_{P}^{G}$ induced from the parabolic
subgroup

$P\coloneqq\left\\{\begin{pmatrix}a&b&c\\\ d&e&f\\\ 0&0&g\end{pmatrix}\in
G\right\\}.$

It is not hard to see that $|G/P|=q^{2}+q+1$ and that
$\psi=\mathbf{1}_{G}+\chi_{qs}$ (cf [12]). Lemma 2.5 hence yields

$\det(\chi_{qs})=(q^{2}+q+1)\cdot(\mathbb{Q}^{\times})^{2}.$

To finish the proof let $q$ be even and regard the characters $\chi_{q^{3}}$.
In both cases these appear in the permutation character $\phi:=1_{B}^{G}$.

For $G=\mathrm{SU}_{3}(q)$ we get $|G/B|=q^{3}+1$ and
$\phi=\mathbf{1}_{G}+\chi_{q^{3}}$ so $\det(\chi_{q^{3}})=q^{3}+1$ by Lemma
2.5.

For $G=\mathrm{SL_{3}}(q)$ we get $|G/B|=(q+1)(q^{2}+q+1)$ and
$\phi=\mathbf{1}_{G}+2\chi_{qs}+\chi_{q^{3}}$. As $\chi_{qs}$ is orthogonally
stable we get

$(q+1)(q^{2}+q+1)\cdot(\mathbb{Q}^{\times})^{2}=\det(\chi_{qs})^{2}\det(\chi_{q^{3}})\cdot(\mathbb{Q}^{\times})^{2}=\det(\chi_{q^{3}})\cdot(\mathbb{Q}^{\times})^{2},$

which finishes the proof. ∎

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