# Probing anomalous $\gamma\gamma\gamma\gamma$ couplings at a future muon
collider

H. Amarkhail
Department of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey
and Department of Physics, Kandahar University, Kandahar 3801, Afghanistan
S.C. İnan
Department of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey
and
A.V. Kisselev
A.A. Logunov Institute for High Energy Physics, NRC “Kurchatov Institute”,
142281, Protvino, Russian Federation Electronic address:
<EMAIL_ADDRESS>address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

We have studied anomalous four-photon couplings in the
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ scattering at a future
muon collider. The collision energies of 3 TeV, 14 TeV, and 100 TeV are
addressed. Both differential and total cross sections versus invariant mass of
the outgoing photons are calculated. The best 95% C.L. exclusion bounds on
anomalous couplings are obtained to be $g_{1}=2.23\times 10^{-8}$ TeV-4 and
$g_{2}=4.22\times 10^{-8}$ TeV-4. They correspond to the muon collision energy
of 100 TeV. The partial-wave unitary constraints on $g_{1}$ and $g_{2}$ are
examined. We have demonstrated that the unitarity is not violated in a region
of the anomalous couplings obtained in the present paper.

##  1 Introduction

In our previous papers, we examined the anomalous quartic gauge couplings
(AQGCs) of the neutral bosons in the processes
$\gamma\gamma\rightarrow\gamma\gamma$ [1] and $\gamma\gamma\rightarrow\gamma
Z$ [2] at the CLIC. It enabled us to probe the anomalous vertices
$\gamma\gamma\gamma\gamma$ and $\gamma\gamma\gamma Z$.

The LHC experimental bounds on neutral AQGCs have been presented by the ATLAS
[11]-[13] and CMS [14, 15] collaborations. Recently, new limits on AQGCs have
been obtained by the CMS and TOTEM collaborations [16] (see also [17]). The
LHC limits on the anomalous triple gauge couplings (ATGCs) were also given
[18]. The sensitivity of AQGCs at future hadron-hadron colliders was
investigated in [20, 21]. Previously a limit on neutral AQGC was obtained at
the LEP using data on the $Z\rightarrow\gamma\gamma\gamma$ decay at the LEP
[19]. At present and future $e^{+}e^{-}$, $e\gamma$, and $\gamma\gamma$
colliders AQGCs were probed in a number of papers [1, 2], [22]-[31].

In the present paper we investigate AQGCs in the process
$\gamma\gamma\rightarrow\gamma\gamma$ at a future high energy muon collider. A
large interest in designing a high energy muon collider is based on its
capability to probe the physics beyond the Standard Model (SM) [32]-[42]. In
particular, it could provide us a determination of the Higgs couplings with a
better accuracy than what is expected at other lepton and hadron future
colliders [43]-[53]. In a number of papers on BSM interactions at the muon
colliders were examined. Among them are the SUSY particles [54], WIMPs
[55]-[57], and dark matter [58], vector boson fusion [59], leptoquarks [60],
lepton flavor violation [61]-[63], vector-like leptons [64], heavy leptons
[65]-[67], and heavy neutrinos [68], Majorana neutrinos [69], top Yukawa
couplings [70], multi-boson processes [71], and physics of the muon $(g-2)$
[72]-[74]. In our recent papers we have probed the warped extra dimension [75]
and axion-like particles [76] at high energy muon colliders. Note also several
papers on searching for AQGCs [77]-[80] and ATGCs [81, 82] at the muon
collider.

##  2 Diphoton production at muon collider

Our goal is to examine the process

$\mu^{+}\mu^{-}\rightarrow\mu^{+}V_{1}V_{2}\mu^{-}\rightarrow\mu^{-}\gamma\gamma\mu^{+}.$
(1)

It is shown in Fig. 1. We do not identify the final state muons, assuming that
the outgoing muons are collinear and thus would escape the detector along the
beampipe.

Without additional cut on invariant mass of the final photons, the dominant SM
background is $\mu^{+}\mu^{-}\rightarrow\gamma\gamma$ [52]. The continuous
background is $\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$. The muon
collider has a reduced initial-state-radiation, contrary to $e^{+}e^{-}$
collider. It means that the tail of the beam energy spread is not large. To
suppress a sizable contribution of the large cross section of
$\mu^{+}\mu^{-}\rightarrow\gamma\gamma$ to a background rate, we impose a
strong upper cut on the diphoton invariant mass,
$m_{\gamma\gamma}<m_{\gamma\gamma,\max}=0.9\sqrt{s}$.

Figure 1: The Feynman diagrams describing diphoton production in the
$\mu^{+}\mu^{-}$ collision via vector boson fusion.

In the equivalent photon approximation (EPA) [83]-[88] the polarized
distributions of photon inside unpolarized fermion beam ($f=\mu^{-}$, in our
case) look like [87]

$\displaystyle
f_{\gamma_{\pm}/f}(x,Q^{2})=f_{\gamma/f}(x,Q^{2})=\frac{\alpha}{4\pi}\frac{1+(1-x)^{2}}{x}\ln\frac{Q^{2}}{m_{\mu}^{2}}\;,$
(2)

where $x=E_{\gamma}/E_{\mu}$ is the ratio of the photon energy $E_{\gamma}$
and energy of the incoming muon $E_{\mu}$, $m_{\mu}$ is the muon mass. Note
that in our case $f=\mu^{-}$. Because of C and P invariance,
$f_{\gamma/\bar{f}}=f_{\gamma/f}$, where $\bar{f}$ denotes anti-fermion
($\bar{f}=\mu^{+}$).

For collisions of massive vector bosons ($W^{\pm}$ and $Z$), the effective $W$
approximation (EWA) is usually used [89, 90]. It allows to treat of massive
vector bosons as partons inside the colliding beams [91]-[99]. In this
approach the $Z$ boson has the following distributions for its transverse
($\pm 1$) and longitudinal ($0$) polarizations [59, 96, 99]

$\displaystyle f_{Z_{\pm}/f}(x,Q^{2})$
$\displaystyle=\frac{\alpha_{Z}}{4\pi}\frac{(g_{V}^{f}\mp
g_{A}^{f})^{2}+(g_{V}^{f}\pm
g_{A}^{f})^{2}(1-x)^{2}}{x}\ln\frac{Q^{2}}{m_{Z}^{2}}\;,$ $\displaystyle
f_{Z_{0}/f}(x,Q^{2})$
$\displaystyle=\frac{\alpha_{Z}}{\pi}\frac{[(g_{V}^{f})^{2}+(g_{A}^{f})^{2}](1-x)}{x}\;,$
(3)

where

$\alpha_{Z}=\frac{\alpha}{(\cos\theta_{W}\sin\theta_{W})^{2}}\;,\ \
g_{V}^{f}=\frac{1}{2}(T_{3}^{f})_{L}-Q^{f}\sin^{2}\theta_{W}\;,\ \
g_{A}^{f}=-\frac{1}{2}(T_{3}^{f})_{L}\;.$ (4)

Again, it is assumed that we deal with the unpolarized fermion beam. The $Z$
boson distributions in anti-fermions are related to those in fermions by CP
relations, $f_{Z_{\pm}/\bar{f}}=f_{Z_{\mp}/f}$,
$f_{Z_{0}/\bar{f}}=f_{Z_{0}/f}$.

The cross section of the process (1) is given by

$d\sigma=\int\limits_{\tau_{\min}}^{\tau_{\max}}\\!\\!d\tau\\!\\!\\!\int\limits_{x_{\min}}^{x_{\max}}\\!\\!\frac{dx}{x}\,\sum_{V_{1},V_{2}}\\!\\!f_{V_{1}/\mu^{+}}(x,Q^{2})f_{V_{2}/\mu^{-}}(\tau/x,Q^{2})\,d\hat{\sigma}(V_{1}V_{2}\rightarrow\gamma\gamma)\;,$
(5)

where $V_{1}$ and $V_{2}$ run over $\gamma,Z_{+},Z_{-},Z_{0}$,

$x_{\max}=1-\frac{m_{\mu}}{E_{\mu}}\;,\
\tau_{\max}=\left(1-\frac{m_{\mu}}{E_{\mu}}\right)^{\\!2},\
x_{\min}=\tau/x_{\max}\;,\ \tau_{\min}=\frac{p_{\bot}^{2}}{E_{\mu}^{2}}\;,$
(6)

and $p_{\bot}$ is the transverse momenta of the outgoing photons. The boson
distributions inside the muon beam, $f_{\gamma/{\mu^{\pm}}}(x,Q^{2})$,
$f_{Z_{\pm}/{\mu^{\pm}}}(x,Q^{2})$, and $f_{Z_{0}/{\mu^{\pm}}}(x,Q^{2})$ are
given by eqs. (2), (2). We take $Q^{2}=\hat{s}$, where
$\sqrt{\hat{s}}=2E_{\mu}\sqrt{\tau}$ is the invariant energy of the VBF
subprocess $V_{1}V_{2}\rightarrow\gamma\gamma$. The differential cross section
in (5) is a sum of squared helicity amplitudes,

$\frac{d\hat{\sigma}}{d\Omega}(V_{1}V_{2}\rightarrow\gamma\gamma)=\frac{1}{64\pi^{2}\hat{s}}\sum_{\lambda_{3},\lambda_{4}}\\!\\!|M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}|^{2},$
(7)

where $\lambda_{1},\lambda_{2}$ are helicity of the incoming bosons, and
$\lambda_{3},\lambda_{4}$ are helicities of the outgoing photons. Remember
that we consider the unpolarized diphoton final state. In its turn, each of
the helicity amplitudes $M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}$ in
(7) is a sum of the anomaly and SM terms,

$M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}=M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}^{\mathrm{anom}}+M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}^{\mathrm{SM}}\;.$
(8)

In what follows, to obtain exclusion bounds for the AQGCs, we assume that
there are no anomalous vertices $Z\gamma\gamma\gamma$ and $ZZ\gamma\gamma$. It
means that only the $\gamma\gamma\rightarrow\gamma\gamma$ subprocess can
contribute to $d\hat{\sigma}(V_{1}V_{2}\rightarrow\gamma\gamma)$ (5) (in other
words, one has to put $V_{1}=V_{2}=\gamma$). Otherwise, one may obtain only
bound on some combination of anomalous couplings corresponding to the
$\gamma\gamma\gamma\gamma$, $Z\gamma\gamma\gamma$, and $ZZ\gamma\gamma$
vertices. Since the distribution of the photon in the unpolarized muon does
not depend on photon’s helicity (2), we have to average 4-photon amplitude in
(7) over helicity states $\lambda_{1},\lambda_{2}$.

To calculate the anomalous helicity amplitudes we use an effective Lagrangian
which contributes to anomalous quartic boson vertices. It can be found, for
instance, in [100]

$\displaystyle\mathcal{L}_{\mathrm{anom}}$
$\displaystyle=\frac{c_{8}}{\Lambda^{4}}B_{\rho\sigma}B^{\rho\sigma}B_{\mu\nu}B^{\mu\nu}+\frac{c_{9}}{\Lambda^{4}}W_{\rho\sigma}^{a}W^{a\rho\sigma}W_{\mu\nu}^{b}W^{b\mu\nu}+\frac{c_{10}}{\Lambda^{4}}W_{\rho\sigma}^{a}W^{b\rho\sigma}W_{\mu\nu}^{a}W^{b\mu\nu}$
$\displaystyle+\frac{c_{11}}{\Lambda^{4}}B_{\rho\sigma}B^{\rho\sigma}W_{\mu\nu}^{a}W^{a\mu\nu}+\frac{c_{13}}{\Lambda^{4}}B_{\rho\sigma}B^{\sigma\nu}B_{\nu\mu}B^{\mu\rho}+\frac{c_{14}}{\Lambda^{4}}W_{\rho\sigma}^{a}W^{a\sigma\nu}W_{\nu\mu}^{b}W^{b\mu\rho}$
$\displaystyle+\frac{c_{15}}{\Lambda^{4}}W_{\rho\sigma}^{a}W^{b\sigma\nu}W_{\nu\mu}^{a}W^{b\mu\rho}+\frac{c_{16}}{\Lambda^{4}}B_{\rho\sigma}B^{\sigma\nu}W_{\nu\mu}^{a}W^{a\mu\rho}.$
(9)

In a broken phase this Lagrangian is expressed in terms of the physical
bosons. We are interested in an effective Lagrangian describing anomalous
$\gamma\gamma\gamma\gamma$ interaction. It is given by

$\mathcal{L}_{\gamma\gamma\gamma\gamma}=g_{1}F_{\mu\nu}F^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}+g_{2}F_{\mu\nu}F^{\mu\rho}F_{\rho\sigma}F^{\sigma\nu},$
(10)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and the
couplings $g_{1}$ and $g_{2}$ are known linear combinations of the
coefficients $c_{8}$, $c_{9}$, $c_{10}$, $c_{11}$, and $c_{13}$, $c_{14}$,
$c_{15}$, $c_{16}$, respectively [100]. Both couplings have dimension TeV-4.
The explicit expressions for
$M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}^{\mathrm{anom}}$ are
collected in Appendix.

In a number of papers bounds on couplings $c_{i}$ (often denoted as $f_{T,i}$)
in EF Lagrangian (2) are searched for. Note, while obtaining bounds on any
coupling, the other couplings were set to zero (see, for instance, [31]). We
prefer to examine couplings of physical fields (like couplings $g_{1},g_{2}$
in the effective Lagrangian (10)). As mentioned above, we will study four-
photon anomalous interaction assuming that there are no quartic interactions
of the photons with one or two $Z$ bosons.

As for the SM contribution to the cross section, all the terms in (5) must be
taken into account ($V_{1},V_{2}=\gamma,Z$). For our numerical calculations we
will use explicit expressions for the SM helicity amplitudes obtained for the
processes $\gamma\gamma\rightarrow\gamma\gamma$ [101]-[104],
$\gamma\gamma\rightarrow\gamma Z$ [105], and $\gamma\gamma\rightarrow ZZ$
[106]. The $W$-loop contribution to the helicity amplitudes is the leading
contribution. Moreover, in the region

$s\sim|t|\sim|u|\gg(m_{W}^{2},m_{Z}^{2})$ (11)

the dominant amplitudes are $M_{++++}$ and $M_{+-+-}$ (for instance, see
[107]), as well as helicity amplitudes related to them by Bose statistics,
crossing symmetry, and parity or time-inversion invariance. Note that in the
high energy region (11) the relative size of the helicity amplitudes for
$\gamma\gamma\rightarrow\gamma\gamma$ and $\gamma\gamma\rightarrow ZZ$ is
$M(\gamma\gamma\rightarrow\gamma\gamma)=(\tan\theta_{W})^{4}M(\gamma\gamma\rightarrow
ZZ)$ [108].

The differential cross sections for different values of the collision energy
$\sqrt{s}$ are presented in Fig. 2 as functions of the diphoton invariant mass
$m_{\gamma\gamma}$. We have used the cut on the rapidity of the outgoing
photons, $|\eta|<2.5$. The curves for two sets of the anomalous couplings
$g_{1}$, $g_{2}$ are shown. In what follows, to suppress the background SM
process $\mu^{+}\mu^{-}\rightarrow\gamma\gamma$, we will impose the cut
$m_{\gamma\gamma}<m_{\gamma\gamma,\max}=0.9\sqrt{s}$. That is why, in Fig. 2
we present the continuous background
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ only. Our cut could not
reduce the total cross section noticeably, since for all three values of the
collision energy the differential cross section decreases rapidly as
$m_{\gamma\gamma}$ approaches $\sqrt{s}$, see Fig. 2. The anomalous part of
the differential cross section becomes to dominate the SM part at large
$m_{\gamma\gamma}$. This effect is more pronounced for $\sqrt{s}=14$ TeV, and
especially for $\sqrt{s}=100$ TeV.

Figure 2: The differential cross sections for the
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ scattering at the future
muon collider versus diphoton invariant mass $m_{\gamma\gamma}$. The center-
of-mass energy is equal to 3 TeV, 14 TeV, and 100 TeV.

In Fig. 3 the total cross section versus _minimal_ value of the parameter
$m_{\gamma\gamma}$ is shown. If $\sqrt{s}=3$ TeV the anomaly part of the total
cross section becomes larger than the SM contribution for
$m_{\gamma\gamma,\min}>0.5$ TeV. But for 14 TeV, and especially for 100 TeV,
the anomaly cross section dominates the SM one almost in the full region of
$m_{\gamma\gamma,\min}$, being several orders of magnitude more at large
$m_{\gamma\gamma,\min}$. Therefore, to get better sensitivity bounds for the
anomalous couplings, we have to use larger values of $m_{\gamma\gamma,\min}$
(see below).

Figure 3: The total cross sections
$\sigma(m_{\gamma\gamma,\min}<m_{\gamma\gamma}<m_{\gamma\gamma,\max})$ for the
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ scattering at the future
muon collider versus minimal value of the diphoton invariant mass
$m_{\gamma\gamma}$.

To obtain exclusion regions, we apply the following formula for the
statistical significance $SS$ [109]

$SS=\sqrt{2[(S-B\,\ln(1+S/B)]}\;,$ (12)

where $S$ is the number of signal events and $B$ is the number of background
events. We define the regions $SS\leqslant 1.645$ as the regions that can be
excluded at the 95% C.L. To reduce the SM background, we take the following
cuts on the invariant mass of the outgoing photons: i) $m_{\gamma\gamma}>0.5$
TeV for $\sqrt{s}=3$ TeV; ii) $m_{\gamma\gamma}>3$ TeV for $\sqrt{s}=14$ TeV;
iii) $m_{\gamma\gamma}>20$ TeV for $\sqrt{s}=100$ TeV. The 95% C.L. exclusion
regions for the anomalous couplings $g_{1}$ and $g_{2}$ are shown in Figs.
4-6. The expected integrated luminosities are taken from [40, 55]. Note that
the total cross section is proportional to the coupling combination
$38g_{1}^{2}+40g_{1}g_{2}+11g_{2}^{2}$, if we use the helicity amplitudes
given in Appendix [1, 115]. It means that the exclusion regions are ellipses
in the $(g_{1}-g_{2})$ plane, and that the cross section is approximately
twice more sensitive to the coupling $g_{1}$ than to $g_{2}$.

Figure 4: The 95% C.L. exclusion region for the couplings $g_{1},g_{2}$ in
unpolarized $\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ scattering
at the muon collider. The inner region of the ellipse is inaccessible. The
collision energy is $\sqrt{s}=3$ TeV, the integrated luminosity is $L=1$ ab-1.
Figure 5: The same as in Fig. 4, but for $\sqrt{s}=14$ TeV and $L=20$ ab-1.
Figure 6: The same as in Fig. 4, but for $\sqrt{s}=100$ TeV and $L=1000$ ab-1.

##  3 Unitarity limits on anomalous couplings

To examine bounds imposed by partial-wave unitarity, we will use the partial-
wave expansion of the helicity amplitude in the center-of-mass system [110],

$\displaystyle
M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}(s,\theta,\varphi)$
$\displaystyle=16\pi\sum_{J}(2J+1)\sqrt{(1+\delta_{\lambda_{1}\lambda_{2}})(1+\delta_{\lambda_{3}\lambda_{4}})}$
$\displaystyle\times\,e^{i(\lambda-\mu)\phi}\,d^{J}_{\lambda\mu}(\theta)\,T^{J}_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}(s)\;,$
(13)

where $\lambda=\lambda_{1}-\lambda_{2}$, $\mu=\lambda_{3}-\lambda_{4}$,
$\theta(\phi)$ is the polar (azimuth) scattering angle, and
$d^{J}_{\lambda\mu}(\theta)$ is the Wigner (small) $d$-function [111]. It
obeys the relation
$d^{J}_{\lambda\mu}(-z)=(-1)^{J-\lambda}d^{J}_{\mu-\lambda}(z)$, where
$z=\cos\theta$. If we choose the plane $(x-z)$ as a scattering plane, then
$\phi=0$ in (3). In what follows, we will need only

$d^{J}_{00}(z)=P_{J}(z)\;,$ (14)

where $P_{J}(z)$ is the Legendre polynomial, and [2]

$d^{J}_{2\pm 2}(z)=(\pm 1)^{J}\\!\left(\frac{1\pm
z}{2}\right)^{\\!2}\\!{}_{2}F_{1}\\!\left(2-J,J+3;1;\frac{1\mp z}{2}\right),$
(15)

where ${}_{2}F_{1}(a,b;c;x)$ is the hypergeometric function [112].

Partial-wave unitarity requires that

$\left|T^{J}_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}(s)\right|\leq
1\;.$ (16)

Using orthogonality of the $d$-functions,

$\int\limits_{-1}^{1}d^{J}_{\lambda\lambda^{\prime}}(z)\,d^{J^{\prime}}_{\lambda\lambda^{\prime}}(z)\,dz=\frac{2}{2J+1}\,\delta_{JJ^{\prime}}\;,$
(17)

we find from (3) that the partial-wave amplitude is given by

$\displaystyle T^{J}_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}(s)$
$\displaystyle=\frac{1}{32\pi}\frac{1}{\sqrt{(1+\delta_{\lambda_{1}\lambda_{2}})(1+\delta_{\lambda_{3}\lambda_{4}})}}\int\limits_{-1}^{1}\\!\\!M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}(s,z)\,d^{J}_{\lambda\mu}(z)\,dz\;.$
(18)

The anomalous helicity amplitudes
$M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}$ are collected in Appendix.

1. Helicity amplitude $M_{++++}$. Then $\lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda_{4}=1$, and $\lambda=\mu=0$. Using explicit expression for $M_{++++}$ (Appendix. Anomalous helicity amplitudes), we find that the partial-wave amplitude with $J=0$ is the only non-zero helicity amplitude,

$T^{J}_{++++}(s)=\frac{(4g_{1}+3g_{2})s^{2}}{128\pi}\\!\int\limits_{-1}^{1}\\!P_{J}(z)\,dz=\frac{(4g_{1}+3g_{2})s^{2}}{64\pi}\delta_{J0}\;.$
(19)

We find from (16), (19) that

$|4g_{1}+3g_{2}|\leq 64\pi s^{-2}\;.$ (20)

2. Helicity amplitude $M_{++--}$. In such a case, $\lambda_{1}=\lambda_{2}=-\lambda_{3}=-\lambda_{4}=1$, and $\lambda=\mu=0$. It follows from (Appendix. Anomalous helicity amplitudes) that $M_{++--}=(4g_{1}+g_{2})s^{2}(3+z^{2})/4$. Then we obtain

$\displaystyle T^{J}_{++--}(s)$
$\displaystyle=\frac{(4g_{1}+g_{2})s^{2}}{256\pi}\int\limits_{-1}^{1}\\!P_{J}(z)(3+z^{2})\,dz$
$\displaystyle=\frac{(4g_{1}+g_{2})s^{2}}{256\pi}\\!\left[\frac{20}{3}\delta_{J0}+\frac{4}{15}\delta_{J2}\right].$
(21)

Thus, non-trivial amplitudes are $T^{0}_{++--}(s)$ and $T^{2}_{++--}(s)$. It
is $T^{0}_{++--}(s)$ that results in more stringent bound on the anomalous
couplings, and we get

$|4g_{1}+g_{2}|\leq(192\pi/5)s^{-2}\;.$ (22)

3. Helicity amplitude $M_{+-+-}$. Then $\lambda_{1}=-\lambda_{2}=\lambda_{3}=-\lambda_{4}=1$, and $\lambda=\mu=2$. Using equation (15) and taking into account that $M_{+-+-}=(4g_{1}+3g_{2})s^{2}[(1+z)/2]^{2}/2$, we get

$\displaystyle T^{J}_{+-+-}(s)$
$\displaystyle=\frac{(4g_{1}+3g_{2})s^{2}}{64\pi}\int\limits_{-1}^{1}\left(\frac{1+z}{2}\right)^{\\!2}\\!d^{J}_{22}(z)\,dz=\frac{(4g_{1}+3g_{2})s^{2}}{32\pi}\int\limits_{0}^{1}(1-x)^{4}$
$\displaystyle\times{}_{2}F_{1}(2-J,J+3;1;x)dx=\frac{3(4g_{1}+3g_{2})s^{2}}{4\pi\Gamma(4+J)\Gamma(3-J)}\;,$
(23)

where $\Gamma(x)$ is the gamma function. To derive (3), eqs. 2.21.1.5 and
2.21.1.6 in [113] were applied. As one can see from (3), only three partial-
waves amplitudes, $T^{0}_{+-+-}(s)$, $T^{1}_{+-+-}(s)$, and $T^{2}_{+-+-}(s)$,
are non-zero. The strongest unitary bound on anomalous couplings comes from
$T^{0}_{+-+-}(s)=(4g_{1}+3g_{2})s^{2}/(16\pi)$. It looks like

$|4g_{1}+3g_{2}|\leq 16\pi s^{-2}\;.$ (24)

4. Helicity amplitude $M_{+--+}$. Since $\lambda_{1}=-\lambda_{2}=-\lambda_{3}=\lambda_{4}=1$, we get $\lambda=2,\,\mu=-2$. After accounting for eqs. (18), (Appendix. Anomalous helicity amplitudes), and (15), it can be shown that we come again to inequality (24).

If the couplings $g_{1}$, $g_{2}$ have the _same_ sign, then we obtain the
following unitary bounds

$|g_{1}|<4\pi s^{-2}\;,\quad|g_{2}|<(16\pi/3)s^{-2}\;.$ (25)

Now let us assume that the couplings $g_{1}$, $g_{2}$ have the _opposite_
sign. It is sufficient to consider the case when $g_{1}>0$, $g_{2}<0$. There
are three possibilities:

1. $4g_{1}>3|g_{2}|$. We get from (22), (24)

$3|g_{2}|<4g_{1}\leq\min\left(3|g_{2}|+16\pi
s^{-2},|g_{2}|+(192\pi/5)s^{-2}\right).$ (26)

It follows from (26) that $g_{1}<(72\pi/5)s^{-2}$ and
$|g_{2}|<(96\pi/5)s^{-2}$.

2. $3|g_{2}|\geq 4g_{1}\geq|g_{2}|$. Then

$|g_{2}|\leq 4g_{1}\leq 3|g_{2}|\leq(708\pi/5)s^{-2}\;.$ (27)

It gives us the upper bounds $g_{1}\leq(107\pi/5)$ and
$|g_{2}|\leq(236\pi/5)s^{-2}$.

3. $4g_{1}<|g_{2}|$. We obtain

$3|g_{2}|-16\pi s^{-2}\leq 4g_{1}<|g_{2}|\;.$ (28)

From (28) we come to the upper limits $g_{1}<2\pi s^{-2}$ and $|g_{2}|<8\pi
s^{-2}$.

As one can see from the inequalities shown above, the most stringent upper
bound on $g_{1}$ looks like

$g_{1}<2\pi s^{-2}\;.$ (29)

So, the unitarity demands that $|g_{1}|$ must be less than $7.8\times 10^{-2}$
TeV-4, $1.64\times 10^{-4}$ TeV-4, and $6.28\times 10^{-8}$ TeV-4, for 3 TeV,
14 TeV, and 100 TeV, respectively. The best bound on $g_{2}$ is achieved when
$g_{1}=0$,

$|g_{2}|\leq(16\pi/3)s^{-2}\;.$ (30)

Correspondingly, the unitary upper limits for $|g_{2}|$ are equal to $0.21$
TeV-4, $4.36\times 10^{-4}$ TeV-4, and $1.68\times 10^{-7}$ TeV-4, for 3 TeV,
14 TeV, and 100 TeV, respectively. Comparing obtained unitary bounds with our
exclusion regions presented in Figs. 4-6, we conclude that the unitary in not
violated in our region of AQGCs. Let us underline that for numerical
calculations of the unitary bounds, in all the above equations, we put $s$ to
be equal to the collision energy of the process
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$. Actually, in these
equations, one has to use the squared invariant energy of the
$\gamma\gamma\rightarrow\gamma\gamma$ subprocess $\hat{s}=sx_{1}x_{2}$ instead
of $s$. As a result, effective unitary bounds are even weaker than those given
above.

##  4 Conclusions

In the present paper we have studied the anomalous quartic couplings of the
$\gamma\gamma\gamma\gamma$ vertex in the unpolarized
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\gamma\gamma\mu^{-}$ scattering at the future
muon collider. The outgoing muons are collinear and thus escape the detector
along the beampipe. To derive the bounds on four-photon couplings, we have
assumed that anomalous couplings of the vertices $Z\gamma\gamma\gamma$ and
$ZZ\gamma\gamma$ are zero. The collision energies of 3 TeV, 14 TeV, and 100
TeV are examined.

Using helicity amplitudes for the light-by-light scattering, we have
calculated both differential and total cross sections. To reduce the
background from the SM $\mu^{+}\mu^{-}\rightarrow\gamma\gamma$ process, the
upper cut on the invariant mass of the outgoing photons is imposed,
$m_{\gamma\gamma}<m_{\gamma\gamma,\max}=0.9\sqrt{s}$. To reduce the continuous
SM background, we have used the lower cuts $m_{\gamma\gamma}>0.5$ TeV, 3 TeV,
and 20 TeV for $\sqrt{s}=3$ TeV, 14 TeV, and 100 TeV, respectively. We have
also imposed the cut on the rapidity of the final photons, $|\eta|<2.5$.

The 95% C.L. exclusion regions for the anomalous four-photon couplings $g_{1}$
and $g_{2}$ are calculated. For $\sqrt{s}=3$ TeV the best bounds appeared to
be $g_{1}=6.9\times 10^{-3}$ TeV-4 (for $g_{2}=0$) and $g_{2}=1.41\times
10^{-2}$ TeV-4 (for $g_{1}=0$). Correspondingly, for $\sqrt{s}=14$ TeV we have
obtained the limits $g_{1}=2.2\times 10^{-6}$ TeV-4, $g_{2}=4.6\times 10^{-6}$
TeV-4. Finally, for $\sqrt{s}=100$ TeV the most stringent bounds on anomalous
couplings are equal to $g_{1}=2.23\times 10^{-8}$ TeV-4 and $g_{2}=4.22\times
10^{-8}$ TeV-4. Thus, new scales of the order of 30 TeV and 100 TeV can be
probed at the future muon collider with the collision energy of 14 TeV and 100
TeV, respectively.

The partial-wave unitary constraints on the anomalous couplings are examined
in detail. We have demonstrated that the unitarity is not violated in the
region of these values of couplings $g_{1}$, $g_{2}$.

The AQGCs can originate from one-loop contribution with a new charged particle
inside the loop. Then one expect that $g_{i}\sim\alpha^{2}/\Lambda^{4}$, where
$\Lambda$ is a mass scale of new physics, and our constraint $g_{2}=4.2\times
10^{-8}$ TeV-4 translates to $\Lambda\simeq 6$ TeV. However, there could be
quite another origin of the AQGCs coming from $s$-channel diagram with a new
neutral mediator. Suppose that its coupling to the SM fields is equal to
$1/f$. Then we expect that $g_{i}\sim 1/(f^{2}\Lambda^{2})$. This new particle
can be, for instance, a KK graviton or radion in the Randall-Sundrum (RS)
model with one extra dimension and warped metric [114]. Taking $1/f\sim 1$
TeV-1 (as in the RS model), we obtain from $g_{2}=4.2\times 10^{-8}$ TeV-4
that $\Lambda\simeq 500$ TeV. Thus, the use of EFT and our cuts are justified
even for the 100 TeV muon collider.

The sensitivity to the $\gamma\gamma\gamma\gamma$ anomalous couplings in
photon collisions at the HL-LHC were estimated to be $g_{1}(g_{2})<1\times
10^{-2}$ TeV-2 [115, 116]. As one can see, it is comparable with the
sensitivity of the 3 TeV muon collider. The sensitivity of the 100 TeV FCC-hh
with integrated luminosity of 3 ab-1 is expected to be $g_{1}(g_{2})<2\times
10^{-4}$ [116], while our bounds on $g_{1}$ and $g_{2}$ for the 100 TeV muon
collider are _four_ orders of magnitude stronger. It demonstrates once again
an advantage of the muon collider over FCC-hh collider in searching for BSM
physics.

Recently the CMS and TOTEM collaborations have presented a first search for
the diphoton production at high mass with tagged protons at 13 TeV [117].
Interpreting the experimental results in an effective dimension-8 extension of
the SM, the following limits at 95% C.L. have been obtained:
$|g_{1}|<2.9\times 10^{-1}$ ($g_{2}=0$) TeV-4, $|g_{2}|<6.0\times 10^{-1}$
TeV-4 ($g_{1}=0$) [117].

## Appendix. Anomalous helicity amplitudes

Here we consider helicity amplitudes for the process
$\gamma\gamma\rightarrow\gamma\gamma$. Bose-Einstein statistics and parity
invariance demand that there exist six independent helicity amplitudes
$M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}$ with $\lambda_{1}=+1$. Only
three of them are nonzero,

$\displaystyle M_{++++}(s,t,u)$
$\displaystyle=\frac{(4g_{1}+3g_{2})}{2}\,s^{2}\;,$ $\displaystyle
M_{++--}(s,t,u)$
$\displaystyle=\frac{(4g_{1}+g_{2})}{2}\,(s^{2}+t^{2}+u^{2})\;,$
$\displaystyle M_{+-+-}(s,t,u)$
$\displaystyle=\frac{(4g_{1}+3g_{2})}{2}\,u^{2}\;,$ (A.1)

where $s$, $t$, $u$ are Mandelstam variables. Other three independent helicity
amplitudes are equal to zero,

$M_{+++-}=M_{++-+}=M_{+-++}=0\;.$ (A.2)

We also have the crossing relations

$\displaystyle M_{+--+}(s,t,u)$
$\displaystyle=M_{+-+-}(s,u,t)=\frac{(4g_{1}+3g_{2})}{2}\,t^{2}\;,$
$\displaystyle M_{+---}(s,t,u)$ $\displaystyle=M_{+-++}(s,u,t)=0\;.$ (A.3)

Equations (Appendix. Anomalous helicity amplitudes)-(Appendix. Anomalous
helicity amplitudes) represent eight helicity amplitudes with
$\lambda_{1}=+1$. The helicity amplitudes with $\lambda_{1}=-1$ can be
obtained from them by the use of the parity relation,

$M_{-\lambda_{2}\lambda_{3}\lambda_{4}}(s,t,u)=M_{+-\lambda_{2}-\lambda_{3}-\lambda_{4}}(s,t,u)\;.$
(A.4)

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# Duplicated Steiner triple systems with self-orthogonal near resolutions

Peter J. Dukes  Mathematics and Statistics, University of Victoria, Victoria,
BC, Canada<EMAIL_ADDRESS>and  Esther R. Lamken  Esther R. Lamken: 773 Colby
Street, San Francisco, CA, USA 94134<EMAIL_ADDRESS>
###### Abstract.

A Steiner triple system, STS$(v)$, is a family of $3$-subsets (blocks) of a
set of $v$ elements such that any two elements occur together in precisely one
block. A collection of triples consisting of two copies of each block of an
STS is called a duplicated Steiner triple system, DSTS. A resolvable (or near
resolvable) DSTS is called self-orthogonal if every pair of distinct classes
in the resolution has at most one block in common. We provide several methods
to construct self-orthogonal near resolvable DSTS and settle the existence of
such designs for all values of $v$ with only four possible exceptions. This
addresses a recent question of Bryant, Davies and Neubecker.

Research of Peter Dukes is supported by NSERC Discovery Grant
RGPIN-2024-03966.

##  1\. Introduction

A _balanced incomplete block design_ (BIBD) is a pair $(V,\mathcal{B})$, where
$\mathcal{B}$ is a collection of subsets (_blocks_) taken from a finite set
$V$ of $v$ elements with the properties:

  1. (1)

Every block contains exactly $k$ elements.

  2. (2)

Every pair of distinct elements of $V$ is contained in precisely $\lambda$
blocks of $\mathcal{B}$.

We denote such a design as a $(v,k,\lambda)$-BIBD. The necessary divisibility
conditions for the existence of a $(v,k,\lambda)$-BIBD are

(1.1) $\displaystyle\lambda(v-1)\equiv 0\pmod{k-1}\text{~{}and}$ (1.2)
$\displaystyle\lambda v(v-1)\equiv 0\pmod{k(k-1)}.$

A _Steiner triple system_ of order $v$, denoted by STS$(v)$, is a
$(v,3,1)$-BIBD. It is well known that STS$(v)$ exist for all $v\equiv 1$ or
$3\pmod{6}$, [12].

A $(v,k,\lambda)$-BIBD $\mathcal{D}=(V,\mathcal{B})$ is called _resolvable_ if
its block collection $\mathcal{B}$ can be partitioned into classes
$R_{1},R_{2},\dots,R_{r}$ (resolution classes) where
$r=\frac{\lambda(v-1)}{k-1}$ such that each element of $V$ is contained in
precisely one block of each class. The resolution classes
$R_{1},R_{2},\dots,R_{r}$, sometimes also called parallel classes, form a
_resolution_ of $\mathcal{D}$. Sometimes the notation $(v,k,\lambda)$-RBIBD is
used for a resolvable BIBD. Necessary conditions for the existence of a
$(v,k,\lambda)$-RBIBD are (1.1) and $v\equiv 0\pmod{k}$. Together, these imply
(1.2).

A $(v,3,1)$-RBIBD or resolvable STS$(v)$ is called a _Kirkman triple system_
and denoted KTS$(v)$. Note that in this case, there are
$|\mathcal{B}|=v(v-1)/6$ blocks and $r=(v-1)/2$ resolution classes where each
class contains $v/3$ blocks. The necessary conditions amount to $v\equiv
3\pmod{6}$. The existence question for KTS$(v)$ was a celebrated open problem
for over 100 years, and known as Kirkman’s schoolgirl problem. In 1971 [17],
Ray-Chaudhuri and Wilson proved that a KTS$(v)$ exists if and only if $v\equiv
3\pmod{6}$. (An earlier and independent proof by Lu [16] was discovered
several years later.)

A $(v,k,\lambda)$-BIBD $\mathcal{D}=(V,\mathcal{B})$ is said to be _near
resolvable_ if $\mathcal{B}$ can be partitioned into classes (which we also
call resolution classes) such that for each element $x\in V$, there is
precisely one class which does not contain $x$ in any of its blocks and each
class contains exactly $v-1$ distinct elements. The number of resolution
classes in the near resolvable setting equals $\lambda v/(k-1)$. It follows
from the above that necessary conditions are $v\equiv 1$ (mod $k$) and
$\lambda=k-1$. For block size $3$, Hanani [11] established the existence of
near resolvable $(v,3,2)$-BIBDs for all $v\equiv 1\pmod{3}$.

Suppose $(V,\mathcal{B})$ is an STS$(v)$. We can easily construct a
$(v,3,2)$-BIBD on $V$ by taking two copies of each block in $\mathcal{B}$. The
resulting block collection can be denoted as $2\mathcal{B}$. We call such a
design a _duplicated_ STS$(v)$ and denote it by DSTS$(v)$. In a recent paper
[5], near resolvable DSTS$(v)$ are constructed for all $v\equiv 1\pmod{6}$,
$v\geq 19$; the cases $v\in\\{7,13\\}$ were shown to be definite exceptions.
Most of the direct constructions in [5] produce designs with an additional
interesting property, which we examine here in more detail.

Consider a resolvable or near resolvable DSTS$(v)$, for $v\equiv 3\pmod{6}$ or
$1\pmod{6}$, respectively. If $|R_{i}\cap R_{j}|\leq 1$ for all $i\neq j$,
then the design is called a _self-orthogonal_ DSTS$(v)$. We caution that there
are other uses of the term ‘orthogonal’ in the context of Steiner triple
systems and other designs, and these can have different meanings; see for
example [6, 8]. The notion of self-orthogonality in [5] and in what follows
here is a property of resolutions. As such, we adopt the standard
abbreviations ‘R’ and ‘NR’ for resolvability and near resolvability,
respectively, and we append $*$ to denote the property that the resolutions
are self-orthogonal. In [5], the NR DSTS$(v)$ constructed for $19\leq v\leq
85$ are all self-orthogonal. The existence of NR∗DSTS$(v)$ for all $v\equiv
1\pmod{6}$, $v\geq 19$, was posed as an open problem. We settle this question
in the affirmative with only a small number of possible exceptions for $v$.

######  Theorem 1.1.

There exists an NR∗DSTS$(v)$ for all $v\equiv 1\pmod{6}$, $v\geq 19$, except
possibly for $v\in\\{115,133,175,259\\}$.

There is a close connection between NR∗DSTSs and other designs with orthogonal
resolutions.

Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be two resolutions of the blocks
of a $(v,k,\lambda)$-BIBD $D$. We say that $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ are _orthogonal_ if $|R_{i}\cap R_{j}^{\prime}|\leq 1$
for all $R_{i}\in\mathcal{R}$ and $R_{j}^{\prime}\in\mathcal{R}^{\prime}$. It
should be noted that the blocks of the design are considered as labeled so
that if a subset of the elements occurs as a block more than once the blocks
are treated as distinct. If $\mathcal{D}$ is a $(v,k,\lambda)$-RBIBD with a
pair of orthogonal resolutions, it is called _doubly resolvable_ and is
denoted by DR$(v,k,\lambda)$-BIBD. If $D$ is an NR$(v,k,\lambda)$-BIBD with a
pair of orthogonal near resolutions, it is called _doubly near resolvable_ and
is denoted by DNR$(v,k,\lambda)$-BIBD.

A great deal of work has been done on the existence of doubly resolvable and
doubly near resolvable designs with block size $3$. Necessary and sufficient
conditions are known for the existence of DR$(v,3,2)$-BIBDs, [14, 10] and
DNR$(v,3,2)$-BIBDS, [13, 10]. DR$(v,3,1)$-BIBDs are equivalent to
KS${}_{3}(v;1,1)$ (Kirkman squares). KS${}_{3}(v;1,1)$ are shown to exist in
[9] with 23 possible exceptions; 11 of these exceptions are constructed in
[4], $v=351$ is constructed in [10], and $v=249$ and $v=357$ are in [1]. So we
have the following updated existence result.

######  Theorem 1.2.

There exists a KS${}_{3}(v;1,1)$ or a DR$((v,3,1)$-BIBD for $v\equiv
3\pmod{6}$ except possibly for $v\in\\{21,141,153,165,177,189,231,261,285\\}$.
Furthermore, there do not exist KS${}_{3}(v;1,1)$ for $v=9$ and $v=15$.

It is easy to use DR$(v,3,1)$-BIBDs with $v\equiv 3\pmod{6}$ to construct
R∗DSTS$(v)$. This construction illustrates the connection between doubly
resolvable designs and self-orthogonal designs.

######  Lemma 1.3.

If there exists a DR$(v,3,1)$-BIBD, there exists an R∗DSTS$(v)$.

  * Proof.

Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be a pair of orthogonal
resolutions for a DR$(v,3,1)$-BIBD. Then $\mathcal{R}\cup\mathcal{R}^{\prime}$
is a set of $2r=v-1$ resolution classes for an R∗DSTS$(v)$. ∎

This immediately gives us the following existence result for R∗DSTS$(v)$.

######  Lemma 1.4.

There exists an R∗DSTS$(v)$ for all $v\equiv 3\pmod{6}$, $v\geq 21$ except
possibly for $v\in\\{21,141,153,165,177,189,231,261,285\\}$.

Our constructions for NR∗DSTS$(v)$ for $v\equiv 1\pmod{6}$ use many of the
techniques and ideas that were used in [9]. In Section 2, we describe direct
constructions for self-orthogonal DSTS based on finite fields. This is similar
to the methods in [18], except that the need for duplicated STS requires extra
conditions in selecting field elements. Our direct constructions simplify and
unify many of the constructions in [5] into an algebraic framework. In Section
3, we provide definitions and existence results for $(1,1;3)$-frames and
describe our main recursive construction using these frames to construct self-
orthogonal near resolvable DSTS. In the following section, the existence of
NR∗DSTS$(v)$ is established for $v\equiv 1\pmod{6}$ with at present four
possible exceptions. Finally, in Section 5, we summarize our results and
discuss generalizations for larger block sizes.

##  2\. Direct constructions

###  2.1. Cyclotomic methods

For a prime power $q$, let $\mathbb{F}_{q}$ denote the finite field of order
$q$ and let $\mathbb{F}_{q}^{\times}$ denote its multiplicative group. Given a
subgroup $G\leq\mathbb{F}_{q}^{\times}$ of even order, a _halfset_ of $G$ is a
subset $H\subset G$ such that $|H|=|G|/2$ and $G=\pm H:=H\cup(-H)$.

Let $q\equiv 7\pmod{12}$ be a prime power, and let $C_{0},\dots,C_{5}$ be the
cosets of index $6$ in the finite field $\mathbb{F}_{q}$. If $\omega$ is a
generator of $\mathbb{F}_{q}$, we have $C_{j}=\\{\omega^{6i+j}:0\leq
i<(q-1)/6\\}$ for each $j=0,1,\dots,5$. Since $q\equiv 7\pmod{12}$, these
cosets have odd size, and it follows that $H=C_{0}$ is a halfset of the
subgroup $G=C_{0}\cup C_{3}$ of index $3$ in $\mathbb{F}_{q}^{\times}$.

Suppose $t,x,y,z$ are elements of $\mathbb{F}_{q}^{\times}$ having the
following properties:

  1. (1)

$x-y,y-z,z-x$ are all nonzero and belong to distinct cosets of $G$; and

  2. (2)

$x,y,z,x+t,y+t,z+t$ are all nonzero and belong to distinct cosets of $H$.

Put $\mathcal{B}=\\{\\{a+hx,a+hy,a+hz\\}:h\in H,a\in\mathbb{F}_{q}\\}$. We
claim that $(\mathbb{F}_{q},\mathcal{B})$ is an NR∗DSTS$(q)$. From condition
(1), it readily follows that $\mathcal{B}$ is the block collection of an
STS$(q)$ on $\mathbb{F}_{q}$. We are interested in the DSTS$(q)$ with block
collection $2\mathcal{B}$. From condition (2), the two blocks $\\{x,y,z\\}$,
$\\{x+t,y+t,z+t\\}$ of $2\mathcal{B}$ develop under multiplication by $H$ into
a partition $R$ of $\mathbb{F}_{q}^{\times}$. The set of $q$ translates of $R$
in $\mathbb{F}_{q}$ thus provide a near resolution of $2\mathcal{B}$. A block
$B$ belongs to two distinct translates of $R$ only if $x+B$ and $y+B$ are both
in $R$ for distinct $x,y\in\mathbb{F}_{q}$. Without loss of generality, we can
take $x=0$, $B=\\{hx,hy,hz\\}$, and $y=ht$ for some $h\in H$. This shows that
the self-orthogonality property holds, and we have an NR∗DSTS$(q)$.

We illustrate the construction in detail for $q=103$, which is a value that
[5] settles for NRDSTS but not for NR∗DSTS.

######  Example 2.1.

Let $q=103$. The element $-3$ has order $17$ in $\mathbb{F}_{q}$, and hence
generates $C_{0}$. If we take $\\{x,y,z\\}=\\{1,4,6\\}$ and $t=20$, conditions
(1) and (2) above can be easily verified. So, this produces an NR∗DSTS$(103)$.
The set of base blocks and translates are given in Table 1.

$\begin{array}[]{ccc}\lx@intercol\hfil\text{base
blocks}\hfil\lx@intercol&\pm\text{translates}\\\
\hline\cr\\{1,4,6\\}&\\{21,24,26\\}&20,83\\\
\\{100,91,85\\}&\\{40,31,25\\}&43,60\\\ \\{9,36,54\\}&\\{86,10,28\\}&77,26\\\
\\{76,98,44\\}&\\{51,73,19\\}&78,25\\\ \\{81,15,74\\}&\\{53,90,46\\}&75,28\\\
\\{66,58,87\\}&\\{47,39,68\\}&84,19\\\ \\{8,32,48\\}&\\{65,89,2\\}&57,46\\\
\\{79,7,62\\}&\\{11,42,97\\}&35,68\\\ \\{72,82,20\\}&\\{70,80,18\\}&101,2\\\
\\{93,63,43\\}&\\{99,69,49\\}&6,97\\\ \\{30,17,77\\}&\\{12,102,59\\}&85,18\\\
\\{13,52,78\\}&\\{67,3,29\\}&54,49\\\ \\{64,50,75\\}&\\{5,94,16\\}&44,59\\\
\\{14,56,84\\}&\\{88,27,55\\}&74,29\\\ \\{61,38,57\\}&\\{45,22,41\\}&87,16\\\
\\{23,92,35\\}&\\{71,37,83\\}&48,55\\\ \\{34,33,101\\}&\\{96,95,60\\}&62,41\\\
\hline\cr\end{array}$ Table 1. Base blocks and translates for an
NR∗DSTS$(103)$.

The preceding construction can, in some cases, be modified to work when
$q\equiv 1\pmod{12}$. Let $q=6\times 2^{e}t+1$, where $t$ is odd and $e$ is a
nonnegative integer. Let $G$ and $H$ be the subgroups of
$\mathbb{F}_{q}^{\times}$ of index $3\times 2^{e}$ and $6\times 2^{e}$,
respectively. We again have $H$ being a halfset of $G$. This time, however, we
need $2^{e}$ distinct quadruples $(x_{i},y_{i},z_{i},t_{i})$,
$i=1,\dots,2^{e}$, such that

  1. (1)

$x_{i}-y_{i},y_{i}-z_{i},z_{i}-x_{i}$ are all nonzero and belong to distinct
cosets of $G$;

  2. (2)

$x_{i},y_{i},z_{i},x_{i}+t_{i},y_{i}+t_{i},z_{i}+t_{i}$ are all nonzero and
belong to distinct cosets of $H$; and

  3. (3)

the $t_{i}$ belong to distinct cosets of $G$.

Similar to before, put $\mathcal{B}=\\{\\{a+hx_{i},a+hy_{i},a+hz_{i}\\}:h\in
H,a\in\mathbb{F}_{q},i=1,\dots,2^{e}\\}$. By condition (3), the values $\pm
ht_{i}$, $h\in H$, $i=1,\dots,2^{e}$ are distinct and we again obtain an
NR∗DSTS$(q)$.

The following result summarizes our application of the above cyclotomic
methods. Table 2 gives more details, including generator elements, starter
blocks, and translates we found for each value of $q=6x+1$. Our results are
limited to what is necessary for the forthcoming proof of the main result,
Theorem 1.1, but the methods work well for values smaller and even larger than
the given range.

######  Lemma 2.2.

There exist NR∗DSTS$(6x+1)$ for all $x\in\\{17,18,21,23,25,26,27,30,33,37,$
$38,46,47,$ $51,58\\}$.

$\begin{array}[]{ccccll}x&q&2^{e}&\omega&\text{starter
block(s)}&\text{translate(s)}\\\ \hline\cr 17&103&1&5&\\{1,4,6\\}&20\\\
18&109&2&6&\\{22,48,56\\}&104\\\ &&&&\\{27,62,86\\}&67\\\
21&127&1&3&\\{1,9,12\\}&50\\\ 23&139&1&2&\\{1,2,4\\}&21\\\
25&151&1&6&\\{1,5,10\\}&2\\\ 26&157&2&5&\\{29,127,151\\}&132\\\
&&&&\\{16,68,107\\}&61\\\ 27&163&1&2&\\{1,2,4\\}&7\\\
30&181&2&2&\\{27,107,153\\}&125\\\ &&&&\\{69,143,161\\}&128\\\
33&199&1&3&\\{1,3,9\\}&41\\\ 37&223&1&3&\\{1,9,12\\}&26\\\
38&229&2&6&\\{116,170,173\\}&94\\\ &&&&\\{45,114,212\\}&15\\\
46&277&2&5&\\{28,114,152\\}&185\\\ &&&&\\{161,182,250\\}&77\\\
47&283&1&3&\\{1,7,23\\}&30\\\ 51&307&1&5&\\{1,15,21\\}&7\\\
58&349&2&2&\\{58,105,124\\}&306\\\ &&&&\\{118,138,333\\}&170\\\
\hline\cr\end{array}$ Table 2. Field constructions for NR∗DSTS$(q)$.

The preceding constructions are less likely to succeed when the largest odd
divisor of $(q-1)/3$ is small. In fact, when $q=6\times 2^{e}+1$ for some $e$,
there does not appear to be a systematic way to choose a halfset of $G$ to
carry out the construction in general. In spite of this, we were able to find
a construction in one of these cases by searching over different initial
blocks and halfsets, getting lucky that one such choice admitted translates
for self-orthogonality.

######  Example 2.3.

Let $q=97$, and use the generator $\omega=5$ of $\mathbb{F}_{q}$. With the
help of a computer, we found a block $B=\\{1,13,17\\}$ and ordered halfset of
$G$, namely

$h=(28,8,67,33,46,70,77,22,63,18,78,50,55,12,52).$

The resulting cyclic STS$(q)$ with base blocks $\\{h_{i}B:i=1,\dots,16\\}$
admits translates

$t=(41,93,79,55,48,45,74,54,34,53,20,1,25,35,70,12)$

such that $\cup_{i=1}^{16}\\{h_{i}B,t_{i}+h_{i}B\\}$ is a self-orthogonal
resolution of the duplicated block collection. This gives an NR∗DSTS$(97)$.

###  2.2. Other constructions

Here, we provide two constructions of NR∗DSTS$(v)$ for non-prime $v$, namely
for $v\in\\{91,121\\}$. The latter value is a prime power, but in this case
the cyclotomic method above has the extra complications that $e=4$ and that
the additive group is not cyclic. Instead, we give cyclic constructions with
multiplier automorphisms.

######  Example 2.4.

Let $v=91$. The element $m=9$ satisfies $m^{3}=1$ in $\mathbb{Z}_{v}$.
Consider the blocks

$\\{1,42,74\\},\\{5,36,39\\},\\{7,33,43\\},\\{48,59,73\\},\\{53,75,82\\}$

and respective translates $79,21,11,13,2$ in $\mathbb{Z}_{v}$. If we develop
these multiplicatively under $1,m,m^{2}$ and then additively, the resulting
near resolvable DSTS$(v)$ has the self-orthogonality property.

######  Example 2.5.

Let $v=121$. The element $m=3$ satisfies $m^{5}=1$ in $\mathbb{Z}_{v}$.
Consider the blocks

$\\{4,10,103\\},\\{14,78,118\\},\\{7,33,43\\},\\{22,96,104\\},\\{71,85,106\\}$

and respective translates $49,27,28,98$ in $\mathbb{Z}_{v}$. If we develop
these multiplicatively under $1,m,m^{2},m^{3}$, $m^{4}$ and then additively,
the resulting near resolvable DSTS$(v)$ has the self-orthogonality property.

##  3\. Frames

Let $T$ denote an integer partition of $v$ and let $K$ be a set of positive
integers. A _group divisible design_ of _type_ $T$ with block sizes in $K$ is
a triple $(V,\mathcal{G},\mathcal{B})$ which satisfies the following
properties.

  1. (1)

$V$ is a set of $v$ points.

  2. (2)

$\mathcal{G}=\\{G_{1},G_{2},\dots,G_{m}\\}$ is a partition of $V$ into groups
so that $T=(|G_{1}|,\dots,|G_{m}|)$.

  3. (3)

$\mathcal{B}\subseteq\cup_{k\in K}\binom{V}{k}$ is a set of blocks meeting
each group in at most one point.

  4. (4)

Every pair of elements from distinct groups occurs in precisely $\lambda$
blocks. ($\lambda$ is called the index.)

We denote such a design as a $GDD(v;K;G_{1},G_{2},\dots,G_{m};0,\lambda)$. It
is convenient to use exponential notation for the type of a GDD; we say a GDD
has type $t_{1}^{u_{1}}t_{2}^{u_{2}}\dots t_{\ell}^{u_{\ell}}$ if there are
$u_{i}$ $G_{j}$’s of cardinality $t_{i}$, $1\leq i\leq l$. A
$GDD(v;K;G_{1},\dots,G_{m};0,1)$ is often denoted as a $K-GDD$ or GDD$(v,K)$
of _type_ $t_{1}^{u_{1}}t_{2}^{u_{2}}\dots t_{\ell}^{u_{\ell}}$. A GDD with a
single block size $k$ is often denoted simply as a $k$-GDD or a GDD$(v,k)$.

We will use two special types of GDDs with index $\lambda=1$ in our recursive
constructions. A _pairwise balanced block design_ or PBD, denoted by
PBD$(v,K)$, is a GDD$(v,K)$ of type $1^{v}$. A _transversal design_ TD$(k,n)$
is a GDD which has block size $k$ and precisely $k$ groups of size $n$, a
$k-GDD$ of type $n^{k}$. In this case, the blocks are transversals of the
partition. It is well known that a TD$(k,n)$ is equivalent to a set of $k-2$
mutually orthogonal latin squares (MOLS) of order $n$. We refer to [3] for
results on the existence of transversal designs.

Let $V$ be a set of $v$ elements. Let $G_{1},G_{2},\dots,G_{m}$ be a partition
of $V$ into $m$ sets. A $\\{G_{1},G_{2},\dots,G_{m}\\}$-_frame_ $F$ with block
size $k$, index $\lambda$, and _latinicity_ $\mu$ is a square array of side
length $t=\lambda v/(\mu(k-1))$ which satisfies the properties listed below.
Let $t_{i}=\lambda|G_{i}|/(\mu(k-1))$, let $g_{k}=\sum_{i=1}^{k}t_{i}$ and
define $g_{0}=0$. We index the rows and columns of $F$ with the elements
$0,1,\dots,\lambda v/(\mu(k-1))-1$.

  1. (1)

Each cell is either empty or contains a $k$-subset of $V$.

  2. (2)

Let $F_{i}$ be the subsquare of $F$ indexed by
$g_{i-1},g_{i-1}+1,\dots,g_{i}-1$. $F_{i}$ is empty for $i=1,2,\dots,m$. That
is, the main diagonal of $F$ consists of empty subsquares of sides
$t_{i}\times t_{i}$ for $i=1,2,\dots,m$.

  3. (3)

Let $x\in\\{g_{i-1},g_{i-1}+1,\dots,g_{i}-1\\}$. Row $x$ of $F$ contains each
element of $V-G_{i}$ $\mu$ times and column $x$ of $F$ contains each element
of $V-G_{i}$ $\mu$ times.

  4. (4)

The blocks obtained from the nonempty cells of $F$ form a
GDD$(v;k;G_{1},G_{2},\dots,G_{m};0,\lambda)$.

The type of a $\\{G_{1},G_{2},\dots,G_{m}\\}$-frame is the type of the
underlying GDD, $\\{|G_{1}|,|G_{2}|,\dots,|G_{m}|\\}$. So, we also use
exponential notation to describe the type of a frame.
A$\\{G_{1},G_{2},\dots,G_{m}\\}$-frame with block size $k$, index $\lambda$,
and latinicity $\mu$ is denoted as a
$(\mu,\lambda;k,\\{G_{1},\dots,G_{m}\\})$-frame or a $(\mu,\lambda;k)$-frame
of type $t_{1}^{u_{1}}t_{2}^{u_{2}}\dots t_{\ell}^{u_{\ell}}$ if there are
$u_{i}\ G_{j}$’s of cardinality $t_{i}$, $1\leq i\leq\ell$. If $|G_{i}|=h$ for
all $i$, the frame $F$ is often called a $(\mu,\lambda;k,m,h)$-frame.

Our main recursive construction uses $(1,1;3)$-frames to construct self-
orthogonal near resolvable DSTSs.

######  Theorem 3.1.

Suppose there exists a $(1,1;3)$-frame with group partition
$\\{G_{1},G_{2},\dots,G_{m}\\}$. If there exists an NR∗DSTS$(|G_{i}|+1)$ for
all $i=1,\dots,m$ then there exists an NR∗DSTS$(\sum_{i=1}^{m}|G_{i}|+1)$.

  * Proof.

Let $V=\cup_{i=1}^{m}G_{i}$. We construct an NR∗DSTS$(v+1)$ on $V\cup\infty$
where $|V|=v$. Let $D_{i}$ denote an NR∗DSTS$(|G_{i}|+1)$ defined on
$G_{i}\cup\infty$ with $|G_{i}|=t_{i}$. $D_{i}$ has $t_{i}+1$ resolution
classes, $P_{i,\infty},P_{i,1},P_{i,2},\dots,P_{i,t_{i}}$ where $P_{i,\infty}$
denotes the resolution class missing the element $\infty$.

Suppose $F$ is a $(1,1;3,\\{G_{1},G_{2},\dots,G_{m}\\})$-frame. $F$ has
$\frac{t_{i}}{2}$ rows, $R_{i,j}$ for $j=1,2,\dots,\frac{t_{i}}{2}$, which are
missing the elements of $G_{i}$; that is, every element of $V\setminus G_{i}$
occurs once in each of these rows. $F$ also has $\frac{t_{i}}{2}$ columns,
$C_{i,j}$ for $j=1,2,\dots,\frac{t_{i}}{2}$, which are missing the elements of
$G_{i}$. Adjoin a resolution class from $D_{i}$ to each of these rows and to
each of these columns. The resulting (near) resolution classes,
$\mathcal{P}_{i,j}$, $j=1,2,\dots,t_{i}$, are each missing precisely one
element of $G_{i}$.

$\mathcal{P}_{i,j}$ | $=$ | $R_{i,j}\cup P_{i,j}$ | for |  $1\leq j\leq\frac{t_{i}}{2}$,
---|---|---|---|---
$\mathcal{P}_{i,\frac{t_{i}}{2}+j}$ | $=$ | $C_{i,j}\cup P_{i.\frac{t_{i}}{2}+j}$ | for |  $1\leq j\leq\frac{t_{i}}{2}$.

Let $\mathcal{P}_{\infty}=\cup_{i=1}^{m}P_{i,\infty}$; every element of $V$
occurs once in this class. Then $\mathcal{P}_{\infty}$ together with
$\mathcal{P}_{i,j}$ for $i=1,2,\dots,m$ and $j=1,2,\dots,t_{i}$ is a set of
$1+\sum_{i=1}^{m}t_{i}$ (near) resolution classes.

First, note that by taking the rows and the columns of the frame $F$, we have
used two copies of the underlying GDD with block size 3 and $\lambda=1$. Since
we are filling in the groups, $G_{i}$, of the GDD with DSTS defined on
$G_{i}\cup\\{\infty\\}$, the resulting design is a DSTS$(v+1)$.

Next we check that the $v+1$ resolution classes are self-orthogonal. Observe
that $C_{i,j}\cap R_{i,k}=\emptyset$ for $1\leq j,k\leq t_{i}$. (The rows and
columns in $F$ which are missing the group $G_{i}$ are disjoint.) Since
$D_{i}$ is an NR∗DSTS, $|\mathcal{P}_{i,j}\cap\mathcal{P}_{i,k}|\leq 1$ for
$j\neq k$ and $|\mathcal{P}_{i,j}\cap\mathcal{P}_{\infty}|\leq 1$. Since $F$
is a $(1,1;3)$-frame, $|\mathcal{P}_{i,j}\cap\mathcal{P}_{\ell,k}|\leq 1$ for
$i\neq\ell$ and $j=1,\dots,t_{i}$, $k=1,\dots,t_{\ell}$. Therefore, we have
constructed an NR∗DSTS$(v+1)$ on $V\cup\\{\infty\\}$. ∎

In order to apply this result, we need some existence results for
$(1,1;3)$-frames. The following two standard recursive constructions for
frames, the singular direct product and the Fundamental Construction, will be
useful.

######  Theorem 3.2 ([9]).

If there exists a $(1,1;3,\\{G_{1},G_{2},\dots,G_{m}\\})$ and a set of 3
mutually orthogonal Latin squares of side $n$, then there exists a
$(1,1;3)$-frame of type $\\{n|G_{i}|\mid i=1,2,\dots,m\\}$.

######  Theorem 3.3 ([9]).

Let $G$ be a GDD$(v;K;G_{1},G_{2},\dots,G_{m};0,1)$. Suppose there exists a
function $w:V\rightarrow\mathbb{Z}^{+}\cup\\{0\\}$ (a weight function) which
has the property that for each block
$b=\\{x_{1},x_{2},\dots,x_{k}\\}\in\mathcal{B}$, there exists a
$(1,1;3)$-frame of type $\\{w(x_{1}),w(x_{2}),\dots,w(x_{k})\\}$. Then there
exists a $(1,1;3)$-frame of type

$\left\\{\sum_{x\in G_{1}}w(x),\sum_{x\in G_{2}}w(x),\dots,\sum_{x\in
G_{m}}w(x)\right\\}.$

[9, Lemma 4.4] describes the connection between Kirkman squares,
KS${}_{3}(v+1;1,1)$, and $(1,1;3)$-frames of type $2^{v}$. The first result
for $(1,1;3)$-frames is a consequence of Theorem 1.2 on the existence of
Kirkman squares (DR$(v,3,1)$-BIBDs).

######  Theorem 3.4.

There exist $(1,1;3)$-frames of type $2^{u}$ for $u\equiv 1\pmod{3}$ and
$u\geq 10$ except possibly for $u\in\\{10,70,76,82,88,94,115,130,142\\}.$

######  Theorem 3.5.

There exist $(1,1;3)$-frames of type $6^{u}$ for $7\leq u\leq 19$,
$u\in\\{31,49,50,56,57,58\\}$, and $u\geq 63$.

  * Proof.

We update the existence result in [2]. $(1,1;3)$-frames of type $6^{u}$ have
been constructed for $u=15,16,17,18$ in [1]. ∎

Several small $(1,1;3)$-frames will also be used in Section 4.

######  Lemma 3.6.

There exist $(1,1;3)$-frames of the following types:

  1. (1)

[7] $4^{7}$; and

  2. (2)

[1] $18^{8}$, $24^{7}18^{1}$, $24^{7}36^{1}$, $30^{7}36^{1}$.

##  4\. Existence

In this section, we construct self-orthogonal near resolvable DSTS$(v)$ for
$v=6x+1$ for $x\geq 3$ with at present four possible exceptions for $x$. Our
main recursive construction uses group divisible designs.

######  Theorem 4.1.

Let $G$ be a GDD$(v;K;G_{1},G_{2},\dots,G_{m};0,1)$. Suppose there exist
$(1,1;3)$-frames of type $6^{k}$ for each $k\in K$ and an NR∗DSTS$(|G_{i}|+1)$
for each $i$, $i=1,2,\dots,m$. Then there exists an NR∗DSTS$(6v+1)$.

  * Proof.

We use the Fundamental Construction for frames, Theorem 3.3 with $w(x)=6$ for
all $x$ and then apply the Basic Frame Construction, Theorem 3.1. ∎

We first use PBDs to construct group divisible designs and then apply Theorem
4.1. This will be our main construction for $v$ large.

######  Theorem 4.2.

If there exists a PBD$(x+1;\\{7,8,9\\})$, then there exists an
NR∗DSTS$(6x+1)$.

  * Proof.

We delete one element of a PBD$(x+1;\\{7,8,9\\})$ to construct a
$\\{7,8,9\\}$-GDD with $|G_{i}|\in\\{6,7,8\\}$ for all $i$. By Theorem 3.5,
there exist $(1,1;3)$-frames of types $6^{k}$ for $k\in\\{7,8,9\\}$, and by
Lemma 4.6 there exist NR∗DSTS$(6m+1)$ for each $m=6,7,8$. So, we can apply
Theorem 4.1 to construct NR∗DSTS$(6x+1)$. ∎

Truncated transversal designs can also be used to construct group divisible
designs. The next result describes several types of group divisible designs
that are useful for the existence of NR∗DSTSs. This is [9, Lemma 5.3].

######  Lemma 4.3 ([9], Truncation of Transversal Designs).

  1. (1)

If there exists a TD$(8,n)$, then there exists a $\\{7,8\\}$-GDD of type
$n^{7}w^{1}$ where $w$ is an integer, $0\leq w\leq n.$

  2. (2)

If there exists a TD$(9,n)$, then there exists a $\\{7,8,9\\}$-GDD of type
$n^{7}w^{1}y^{1}$ where $w$ and $y$ are integers, $0\leq w,y\leq n.$

  3. (3)

If there exists a TD$(10,n)$, then there exists a $\\{7,8,9,10\\}$-GDD of type
$(n-1)^{8}w^{1}y^{1}$ where $w$ and $y$ are integers, $0\leq w,y\leq n-1.$

  4. (4)

There exists a $\\{7,8,9,w,19\\}$-GDD of type $7^{19-w}8^{w}y^{1}$ whenever
$0\leq w\leq 19$ and $0\leq y\leq 18.$

In order to apply Theorem 4.2, we need the existence of PBD$(v;\\{7,8,9\\})$s.
Let $E_{789}=[10,48]\cup[51,55]\cup[59,62]$ and
$X_{789}=[93,111]\cup[116,118]\cup\\{132\\}\cup[138,168]\cup[170,174]\cup[180,216]\cup[219,223]\cup[228,230]\cup[242,258]\cup[261,279]\cup[283,286]\cup[298,300]\cup[303,307]\cup[311,335]\cup[339,342]$
where the notation $[x,y]$ is used to denote the set of integers no smaller
than $x$ and no larger than $y$. We use the existence result, [9, Theorem
5.6]; it updates the existence result in [15]

######  Theorem 4.4 ([9]).

For any integer $v\geq 10$, there is a $PBD(v;\\{7,8,9\\})$ except possibly
when $v$ is in $X_{789}$, and definitely when $v$ is in $E_{789}$.

One other recursive construction is useful for several small cases. It follows
immediately from Theorem 3.2 and Theorem 3.1.

######  Theorem 4.5.

If there exists a $(1,1;3)$-frame of type $t^{m}$, three mutually orthogonal
Latin squares of order $n$, and an NR∗DSTS$(tn+1)$, then there exists an
NR∗DSTS$(tmn+1)$.

We are now in a position to prove our main result on the existence of
NR∗DSTS$(v)$ for $v\equiv 1\pmod{6}$. We recall that these do not exist [5]
for $v\in\\{7,13\\}$.

######  Lemma 4.6.

There exist NR∗DSTS$(v)$ for all $v=6x+1$, $3\leq x\leq 52$, except possibly
for $x\in\\{19,$ $22,29,43\\}$.

  * Proof.

NR∗DSTS$(6x+1)$ for $3\leq x\leq 14$ are given in [5]. We use Lemma 2.2 for
several cases with $15\leq x\leq 52$. Constructions for the remaining cases
are described in Table 3. ∎

  * Proof of Theorem 1.1.

Lemma 4.6 establishes the existence of NR∗DSTS$(6x+1)$ for $3\leq x\leq 52$.
For $53\leq x\leq 369$, all but one of these values is done using the group
divisible design construction, Theorem 4.1. These constructions are described
in detail in Table 4. In each case, we make use of the existence of the small
designs constructed in Lemma 4.6 and the existence of the necessary frames of
type $6^{k}$, Theorem 3.5.

By Theorems 4.2 and 4.4, there exist PBD$(x+1;\\{7,8,9\\})$ and hence
NR∗DSTS$(6x+1)$ for all $x\geq 342$. ∎

##  5\. Concluding Remarks

We have shown that there exist self-orthogonal near resolvable DSTS$(v)$ for
all $v\equiv 1\pmod{6}$, $v\geq 19$, with four possible exceptions. The
largest exception could be constructed using a $(1,1;3)$-frame of type
$18^{13}24^{1}$ if it were found. The remaining values are non-prime-powers
divisible by either $5$ or $7$, and may require new ideas to obtain a
construction.

One could consider a generalization of this work to constructing near
resolvable $(v,k,k-1)$-BIBDs via $k-1$ copies of a $(v,k,1)$-BIBD with
$v\equiv 1\pmod{k(k-1)}$. Let $(V,\mathcal{B})$ be a $(v,k,1)$-BIBD with
$v\equiv 1\pmod{k(k-1)}$. We denote a self-orthogonal NR$(v,k,1)$-BIBD defined
on $V$ with block collection $(k-1)\mathcal{B}$ by NR∗ $(v,k,k-1)$-BIBD. The
direct cyclotomic construction of Section 2 generalizes to larger block size
and can be used to find examples. Here is an example of a NR∗ $(v,4,3)$-BIBD
obtained as three disjoint translates of a cyclic $(v,4,1)$-BIBD.

######  Example 5.1.

Let $q=37$ and consider the cosets $H=C_{0},\dots,C_{11}$ of index $12$ in
$\mathbb{F}_{q}$. The element $\omega=2$ is a generator, so we have
$H=\\{2^{0},2^{12},2^{24}\\}=\\{1,26,10\\}$. The starter block
$B=\\{1,4,6,15\\}$ has its 12 ordered differences in distinct cosets, and
moreover $B\cup(10+B)\cup(18+B)$ intersects each coset $C_{j}$ exactly once.
Letting $\mathcal{B}=\\{a+hB:h\in H,a\in\mathbb{F}_{q}\\}$, we obtain blocks
of a $(37,4,1)$-BIBD. The blocks $\\{hB,h(B+10),h(B+18):h\in H\\}$ provide a
starter for a near resolvable $(37,4,3)$-BIBD. Since $\pm 10$, $\pm 18$ (the
two translates), and $\pm 8$ (their difference) are in distinct cosets, the
tripled block collection $3\mathcal{B}$ is an NR∗ $(37,4,3)$-BIBD.

Computer searches found instances of Example 5.1 for $q=109$ and $q=157$ but,
curiously, not for $q=61$.

NR∗ $(v,k,k-1)$-BIBDs are PBD-closed, [19]. To see this, we construct a
$v\times v$ array from an NR∗ $(v,k,k-1)$-BIBD as follows. Index the rows and
columns of the array with the (near) resolution classes
$R_{1},R_{2},\dots,R_{v}$ ordered so that $R_{i}$ is missing element $i$ of
$V$. For $i\neq j$, place the block from $R_{i}\cap R_{j}$ in cell $(i,j)$. If
$R_{i}\cap R_{j}=\emptyset$, the cell is left empty. The diagonal is also left
empty. The resulting square array displays the resolution classes in the
columns and in the rows.

######  Lemma 5.2.

If there exists a PBD$(v,M)$ and an NR∗ $(m,k,k-1)$-BIBD for each $m\in M$,
then there exists an NR∗ $(v,k,k-1)$-BIBD.

  * Proof.

Construct a $v\times v$ array by replacing each block of size $m$ with the
$m\times m$ array obtained from an NR∗ $(m,k,k-1)$-BIBD. It is straightforward
to check that the resulting array displays a set of self-orthogonal
resolutions for a NR∗ $(v,k,k-1)$-BIBD. ∎

Although PBDs are not useful for explicit constructions here due to the large
block sizes required, Lemma 5.2 can be used to establish asymptotic existence
results. For example, let $M=\\{37,157\\}$. There exist PBD$(v,M)$ for all
sufficiently large $v\equiv 1\pmod{12}$, [19]. Since NR∗ $(m,4,3)$-BIBDs exist
for each $m\in M$, there exist NR∗ $(v,4,3)$-BIBDs for all sufficiently large
$v\equiv 1\pmod{12}$.

For general values of $k$, the existence of NR∗ $(v,k,k-1)$-BIBDs is an
interesting topic left for future work.

## References

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## Appendix

Two tables are given for the construction of small designs needed for our
proof.

Table 3. Constructions for NR∗DSTS$(6x+1)$ for $3\leq x\leq 52$ $x$ | $6x+1$ | Construction | Parameters
---|---|---|---
$3,4,\dots,14$ | $19,25,\dots,85$ | [5] | Direct
15 | 91 | 2.4 | Direct
16 | 97 | 2.3 | Direct
17 | 103 | 2.2 | Direct
18 | 109 | 2.2 | Direct
19 | 115 | ?? |
20 | 121 | 2.5 | Direct
21 | 127 | 2.2 | Direct
22 | 133 | ?? |
23 | 139 | 2.2 | Direct
24 | 145 | 3.1 |  $18^{8}$-frame
25 | 151 | 2.2 | Direct
26 | 157 | 2.2 | Direct
27 | 163 | 2.2 | Direct
28 | 169 | 4.5 |  $6^{7}$-frame, $n=4$
29 | 175 | ?? |
30 | 181 | 2.2 | Direct
31 | 187 | 3.1 |  $24^{7}18^{1}$-frame
32 | 193 | 4.5 |  $6^{8}$-frame, $n=4$
33 | 199 | 2.2 | Direct
34 | 205 | 3.1 |  $24^{7}36^{1}$-frame
35 | 211 | 4.5 |  $6^{7}$-frame, $n=5$
36 | 217 | 4.5 |  $6^{9}$-frame, $n=4$
37 | 223 | 2.2 | Direct
38 | 229 | 2.2 | Direct
39 | 235 | 4.5 |  $2^{13}$-frame, $n=9$
40 | 241 | 4.5 |  $6^{10}$-frame, $n=4$
41 | 247 | 3.1 |  $30^{7}36^{1}$-frame
42 | 253 | 4.5 |  $4^{7}$-frame, $n=9$
43 | 259 | ?? |
44 | 265 | 4.5 |  $6^{11}$-frame, $n=4$
45 | 271 | 4.5 |  $6^{9}$-frame, $n=5$
46 | 277 | 2.2 | Direct
47 | 283 | 2.2 | Direct
48 | 289 | 4.5 |  $6^{12}$-frame, $n=4$
49 | 295 | 4.5 |  $6^{7}$-frame, $n=7$
50 | 301 | 4.5 |  $6^{10}$-frame, $n=5$
51 | 307 | 2.2 | Direct
52 | 313 | 4.5 |  $6^{13}$-frame, $n=4$
Table 4. Constructions for NR∗DSTS$(6x+1)$ for $53\leq x\leq 369$ $x$ | Construction | Parameters | Ingredient design
---|---|---|---
$53\leq x\leq 56$ |  4.1, 4.3(1) |  $n=7$, $4\leq w\leq 7$ |  $\\{7,8\\}$-GDD of type $7^{7}w^{1}$
$57$ |  4.2, 4.4 |  |  PBD$(x+1;\\{7,8,9\\})$
$58$ | 2.2 | Direct |
$59\leq x\leq 61$ |  4.1, 4.3(2) |  $n=8$, $w=0$, $3\leq y\leq 5$ |  $\\{7,8,9\\}$-GDD of type $8^{7}w^{1}y^{1}$
$62\leq x\leq 91$ |  4.2, 4.4 |  |  PBD$(x+1;\\{7,8,9\\})$
$83\leq x\leq 100$ |  4.1, 4.3(3) |  $n=11$, $3\leq w,y\leq 10$ |  $\\{7,8,9,10\\}$-GDD of type $10^{8}w^{1}y^{1}$
$99\leq x\leq 120$ |  4.1, 4.3(3) |  $n=13$, $3\leq w,y\leq 12$ |  $\\{7,8,9,10\\}$-GDD of type $12^{8}w^{1}y^{1}$
$118\leq x\leq 130$ |  4.2, 4.4 |  |  PBD$(x+1;\\{7,8,9\\})$
131 | 4.1 | [9, Lemma 5.4] |  $\\{7,8,9\\}$-GDD of type $8^{14}12^{1}7^{1}$
$132\leq x\leq 136$ |  4.2, 4.4 |  |  PBD$(x+1;\\{7,8,9\\})$
$137\leq x\leq 148$ |  4.1, 4.3(4) |  $w=0$, $4\leq y\leq 15$ |  $\\{7,8,9,19\\}$-GDD of type $7^{19}y^{1}$
$144\leq x\leq 155$ |  4.1, 4.3(4) |  $w=7$, $4\leq y\leq 15$ |  $\\{7,8,9,19\\}$-GDD of type $7^{12}8^{7}y^{1}$
$150\leq x\leq 161$ |  4.1, 4.3(4) |  $w=13$, $4\leq y\leq 15$ |  $\\{7,8,9,13,19\\}$-GDD of type $7^{6}8^{13}y^{1}$
$162$ |  4.1, 4.3(4) |  $w=15$, $y=14$ |  $\\{7,8,9,15,19\\}$-GDD of type $7^{4}8^{15}y^{1}$
$163$ |  4.1, 4.3(4) |  $w=15$, $y=15$ |  $\\{7,8,9,15,19\\}$-GDD of type $7^{4}8^{15}y^{1}$
$164\leq x\leq 205$ |  4.1, 4.3(2) |  $n=23$, $3\leq w+y\leq 44$ |  $\\{7,8,9\\}$-GDD of type $23^{7}w^{1}y^{1}$
|  | $w,y\in\\{3,\dots,23\\}\setminus\\{19,22\\}$ |
$178\leq x\leq 225$ |  4.1, 4.3(2) |  $n=25$, $3\leq w+y\leq 50$ |  $\\{7,8,9\\}$-GDD of type $25^{7}w^{1}y^{1}$
|  | $w,y\in\\{3,\dots 25\\}\setminus\\{19,22\\}$ |
$220\leq x\leq 279$ |  4.1, 4.3(2) |  $n=31$, $3\leq w+y\leq 62$ |  $\\{7,8,9\\}$-GDD of type $31^{7}w^{1}y^{1}$
|  | $w,y\in\\{3,\dots,31\\}\setminus\\{19,22,29\\}$ |
$262\leq x\leq 333$ |  4.1, 4.3(2) |  $n=37$, $3\leq w+y\leq 74$ |  $\\{7,8,9\\}$-GDD of type $37^{7}w^{1}y^{1}$
|  | $w,y\in\\{3,\dots,37\\}\setminus\\{19,22,29\\}$ |
$290\leq x\leq 369$ |  4.1, 4.3(2) |  $n=41$, $3\leq w+y\leq 82$ |  $\\{7,8,9\\}$-GDD of type $41^{7}w^{1}y^{1}$
|  | $w,y\in\\{3,\dots,41\\}\setminus\\{19,22,29\\}$ |