The LHC is actively searching for  light pseudoscalar particles.  For a light flavon, present sensitivities  of the LHC  are given in table  <ref>. Using these sensitivities given in table  <ref>, the estimated sensitivities for the HL-LHC, HE-LHC, and a 100 TeV collider are given in table <ref>.  The benchmark predictions for the processes given in table <ref>, for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries, are given in tables  <ref>- <ref> for  $m_a= 20,60$ GeV for the soft symmetry-breaking scenario.

2c|$\mathcal{L}[fb^{-1}]$ [References]     2c|ATLAS 13 TeV     2cCMS 13 TeV

$m_{a}$ [GeV]      cATLAS     c|CMS     c20     c|60     c20     c60

$\tau \tau$ [pb]
$138$ [142]                      $4$

$\gamma \gamma$ [pb]    $138$ [143, 144]      [145]      $6$     $7$

Current limits of $\sigma \times BR$ at 13 TeV LHC by ATLAS and CMS in low mass resonance searches for inclusive flavon production channels.

2c|HL-LHC [14 TeV, $3~\iab$]      2c|HE-LHC [27 TeV, $15~\iab$]     2c100 TeV, $30~\iab$

$m_{a}$ [GeV]      20      60      20      60      20      60

$\tau \tau$ [pb]          $0.9$         $0.5$          $0.6$

$\gamma \gamma$ [pb]     $1.3$      $1.5$      $0.7$     $0.8$      $0.8$     $0.9$

Estimated reach ($\sigma \times BR$) of the HL-LHC, HE-LHC and a 100 TeV collider for low flavon mass ($m_a$) in inclusive flavon production channels.

We observe from table <ref> that the HL-LHC provides access only to probe the $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry through the $\tau \tau$ channel for the mass $m_a= 60$ GeV.  The situation changes greatly as we approach the HE-LHC and a 100 TeV hadron collider, as shown in tables <ref> and <ref>.  Now, the  $\tau \tau$ channel is accessible for all $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries.  The $\gamma \gamma$ channel and the flavon mass $m_a= 20$ GeV remain beyond the reach of the HL-LHC, HE-LHC and a 100 TeV collider for all flavour symmetries.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$\tau \tau$ [pb]          $4.7\e{-3}$     $0.58$      $1.1\e{-2}$     $0.62$     $2.2\e{-2}$     $0.69$      $7.7\e{-3}$     $9.62$

$\gamma \gamma$ [pb]          $5.6\e{-4}$     $1\e{-3}$      $1.5\e{-4}$     $9.9\e{-5}$     $1.1\e{-4}$     $5.8\e{-5}$     $9\e{-5}$     $1.7\e{-2}$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 14 TeV HL-LHC  , assuming $f= 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$\tau \tau$ [pb]          $8\e{-3}$     $1.06$      $2.2\e{-2}$     $1.38$     $4.2\e{-2}$     $1.55$      $1.5\e{-2}$     $21.78$

$\gamma \gamma$ [pb]          $9.4\e{-4}$     $1.8\e{-3}$      $2.9\e{-4}$     $2.2\e{-4}$     $2.2\e{-4}$     $1.3\e{-4}$     $1.7\e{-4}$     $3.8\e{-2}$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 27 TeV HE-LHC, assuming $f= 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$\tau \tau$ [pb]          $2.2\e{-2}$     $3.29$      $7.5\e{-2}$     $5.72$     $0.15$     $6.43$      $5.1\e{-2}$     $92.23$

$\gamma \gamma$ [pb]          $2.6\e{-3}$     $5.7\e{-3}$      $9.5\e{-4}$     $9.1\e{-4}$     $7.5\e{-4}$     $5.3\e{-4}$     $5.5\e{-4}$     $0.16$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at a 100 TeV collider, assuming $f= 500$ GeV.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$\tau \tau$ [pb]          $1.9\e{-7}$     $5.1\e{-4}$      $1.1\e{-11}$     $2\e{-6}$     $2.4\e{-13}$     $3.3\e{-7}$

$\gamma \gamma$ [pb]          $3.8\e{-7}$     $8.8\e{-7}$      $4.5\e{-10}$     $3.2\e{-10}$     $3.8\e{-11}$     $2.7\e{-11}$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at the 14 TeV HL-LHC.

In the symmetry-conserving case, we observe from figures  <ref>  and  <ref> that the cross-section can be relatively significant only for a low flavon mass.  Therefore, we investigate the symmetry-conserving scenario only for $m_a= 20, 60$ GeV.  We show benchmark predictions of symmetry-conserving scenario for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries in tables  <ref>- <ref> for  $m_a= 20,60$ GeV. It turns out that the inclusive processes in this scenario are beyond the reach of the HL-LHC, HE-LHC, and even a 100 TeV collider.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$\tau \tau$ [pb]          $2\e{-7}$     $9.2\e{-4}$      $1.8\e{-11}$     $4.5\e{-6}$     $4\e{-13}$     $7.3\e{-7}$

$\gamma \gamma$ [pb]          $6.3\e{-7}$     $1.6\e{-6}$      $8.8\e{-10}$     $7.2\e{-10}$     $7.4\e{-11}$     $6\e{-11}$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at the 27 TeV HE-LHC.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$\tau \tau$ [pb]          $2.7\e{-10}$     $2.8\e{-3}$      $4.9\e{-11}$     $1.9\e{-5}$     $1.1\e{-12}$     $3\e{-6}$

$\gamma \gamma$ [pb]          $1.7\e{-6}$     $4.9\e{-6}$      $2.8\e{-9}$     $2.9\e{-9}$     $2.4\e{-10}$     $2.5\e{-10}$

Benchmark points for different $\mathcal{Z}_N \times \mathcal{Z}_{M}$ flavour symmetries for inclusive flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at a 100 TeV collider.

§.§.§ Associative production

We notice that there are other channels through which flavon can be produced, as shown in figure <ref>.  For instance, the following associated production processes can be used to produce the flavon,
bg →b a,  ug, c g  →t a,  bb̅ a →bb̅ τ^+ τ^-,  tt̅ a →tt̅  tt̅,
where the first process undergoes through the  flavour-diagonal coupling of the flavon to the $b\bar{b}$, and the second process occurs through the flavour-violating flavon-quark coupling.

Feynman diagrams for the production of the flavon through different associative channels.

Production cross-sections of the flavon of  $\mathcal{Z}_2 \times \mathcal{Z}_{5}$, and $\mathcal{Z}_2 \times \mathcal{Z}_{9}$ flavour symmetries with respect to its mass through different channels for 14 TeV HL-LHC, 27 TeV HE-LHC and future 100 TeV collider. The solid lines represents the production cross section along the boundaries of parameter space allowed by the observable $R_{\mu \mu}$ for soft symmetry-breaking scenario, while the dashed lines correspond to that in the symmetry-conserving scenario along the allowed parameter space by the observable $BR(K_L\rightarrow \mu \mu)_{SD}$.

The cross-sections of different associative production channels are also shown in figures <ref> and <ref>.  We observe that the production cross sections are, in general, smaller for the soft symmetry-breaking as well as symmetry-conserving scenarios, and can be relatively large only for low flavon mass.

The process $t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ can occur only for the  $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry where  the flavour-diagonal coupling of the flavon to top quarks is allowed.  This results in a relatively large cross-section involving  $ t  \bar{t} $ channels. Thus, the $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry is very special in this sense.

As noticed in the case of inclusive production of the heavy flavon mass, we observe from figures <ref> and <ref> that the associative production cross-sections of the flavon of different $\mathcal{Z}_{\rm N}\times \mathcal{Z}_{\rm M}$ flavour symmetries for the VEV $f =500$ GeV for the soft symmetry-breaking scenario can be sufficiently large.   Therefore, we continue to use this choice even for the associative production  of the heavy flavon of different $\mathcal{Z}_{\rm N}\times \mathcal{Z}_{\rm M}$ flavour symmetries.

2c|$\mathcal{L}[fb^{-1}]$ [References]     2c|ATLAS 13 TeV     2cCMS 13 TeV

$m_{a}$ [GeV]      cATLAS     c|CMS     c500     c|1000     c500     c1000

$t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ [pb]      $139$ [146]      $137$ [147]       $1\e{-2}$    xx  $6\e{-3}$        $2\e{-2}$    xx

$gg \rightarrow a b \bar{b} \rightarrow \tau \tau b \bar{b} $ [pb]
$36.1$ [132]     $35.9$ [133]      $8\e{-2}$      $9\e{-3}$       $6\e{-2}$      $1\e{-2}$

$gb \rightarrow a b  \rightarrow \tau \tau b  $ [pb] )       $36.1$ [132]     $35.9$ [133]      $3\e{-2}$     $4\e{-3}$      $3\e{-2}$     $4\e{-3}$

Current limits of $\sigma \times BR$ at 13 TeV LHC by ATLAS and CMS in resonance searches for associative flavon production channels.

The present reach of the LHC for the associative production channels is shown in table  <ref>.   We show the sensitivities of these modes at the HL-LHC, HE-LHC,  and a 100 TeV collider in table  <ref>.  Our benchmark predictions for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetries, are given in tables  <ref>- <ref> for  the soft symmetry-breaking scenario.

Production cross-sections of the flavon of  $\mathcal{Z}_2 \times \mathcal{Z}_{11}$, and $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetries with respect to its mass through different channels for 14 TeV HL-LHC, 27 TeV HE-LHC and future 100 TeV collider. The solid lines represents the production cross section along the boundaries of parameter space allowed by the observable $R_{\mu \mu}$ for soft symmetry-breaking scenario, while the dashed lines correspond to that in the symmetry-conserving scenario along the allowed parameter space by the observable $BR(K_L\rightarrow \mu \mu)_{SD}$.

2c|HL-LHC [14 TeV, $3~\iab$]      2c|HE-LHC [27 TeV, $15~\iab$]     2c100 TeV, $30~\iab$

$m_{a}$ [GeV]      500      1000      500      1000      500      1000

$t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ [pb]     $2\e{-3}$     $1\e{-3}$     $3\e{-3}$     $2\e{-3}$     $1\e{-2}$     $9\e{-3}$

$gg \rightarrow a b \bar{b} \rightarrow \tau \tau b \bar{b} $ [pb]     $7\e{-3}$      $1\e{-3}$      $5\e{-3}$     $8\e{-4}$      $8\e{-3}$     $1\e{-3}$

$gb \rightarrow a b  \rightarrow \tau \tau b  $ [pb]        $3\e{-3}$     $5\e{-4}$      $2\e{-3}$      $3\e{-4}$       $4\e{-3}$      $5\e{-4}$

Estimated reach ($\sigma \times BR$) of HL-LHC, HE-LHC and the 100 TeV collider for high flavon mass ($m_a$) in associative flavon production channels.

We observe from table <ref> that only the mode $t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ is within the reach of the HL-LHC for the flavon mass $m_a = 500$ GeV for the $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry.  This scenario improves slightly at the HE-LHC and at a 100 TeV collider for  the $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry such that the flavon mass $m_a =1000$ GeV becomes also accessible. The symmetry-conserving scenario remains beyond the reach of the  HL-LHC, HE-LHC and a 100 TeV collider  for the  associative production modes. Therefore, we do not discuss it anymore in this case.

Production cross-sections of the flavon of  $\mathcal{Z}_2 \times \mathcal{Z}_{5}$, and $\mathcal{Z}_2 \times \mathcal{Z}_{9}$ flavour symmetries with respect to its mass through different channels for 14 TeV HL-LHC, 27 TeV HE-LHC, and future 100 TeV hadron collider, where the flavon VEV $f=500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ [pb]                                       $1.6\e{-3}$     $1.6\e{-4}$

$gg \rightarrow a b \bar{b} \rightarrow \tau \tau b \bar{b} $ [pb]          $6\e{-7}$     $1.8\e{-8}$      $1\e{-4}$     $3.2\e{-6}$     $3.4\e{-5}$     $8.9\e{-7}$     $1.6\e{-6}$     $3.2\e{-8}$

$gb \rightarrow a b  \rightarrow \tau \tau b  $ [pb]          $1.7\e{-7}$     $5.6\e{-9}$      $2.9\e{-5}$     $9.7\e{-7}$     $9.4\e{-6}$     $2.7\e{-7}$     $4.5\e{-7}$     $9.7\e{-9}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for associative flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking at the 14 TeV HL-LHC, with $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ [pb]                                       $1.5\e{-2}$     $2.3\e{-3}$

$gg \rightarrow a b \bar{b} \rightarrow \tau \tau b \bar{b} $ [pb]          $3.3\e{-6}$     $1.6\e{-7}$      $5.8\e{-4}$     $2.8\e{-5}$     $1.9\e{-4}$     $7.9\e{-6}$     $9\e{-6}$     $2.8\e{-7}$

$gb \rightarrow a b  \rightarrow \tau \tau b  $ [pb]          $9.3\e{-7}$     $5.1\e{-8}$      $1.6\e{-4}$     $8.8\e{-6}$     $5.3\e{-5}$     $2.5\e{-6}$     $2.5\e{-6}$     $8.8\e{-8}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for associative flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking at the 27 TeV HE-LHC, with $f = 500$ GeV.

Production cross-sections of the flavon of  $\mathcal{Z}_2 \times \mathcal{Z}_{11}$, and $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetries with respect to its mass through different channels for 14 TeV HL-LHC, 27 TeV HE-LHC, and future 100 TeV hadron collider, where the flavon VEV $f=500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $ [pb]                                       $0.41$     $9.6\e{-2}$

$gg \rightarrow a b \bar{b} \rightarrow \tau \tau b \bar{b} $ [pb]          $4.7\e{-5}$     $4.0\e{-6}$      $8.2\e{-3}$     $6.9\e{-4}$     $2.7\e{-3}$     $1.9\e{-4}$     $1.3\e{-4}$     $6.8\e{-6}$

$gb \rightarrow a b  \rightarrow \tau \tau b  $ [pb]          $1.34\e{-5}$     $1.23\e{-6}$      $2.3\e{-3}$     $2.1\e{-4}$     $7.5\e{-4}$     $6.0\e{-5}$     $3.6\e{-5}$     $2.1\e{-6}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for associative flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at a 100 TeV collider, assuming $f = 500$ GeV.

§.§.§ Di-flavon production

In this section, we discuss the production of di-flavon, which is an important mode to explore the flavon physics of different  $\mathcal{Z}_{\rm N}\times \mathcal{Z}_{\rm M}$ flavour symmetries.  We show the present LHC sensitivities of di-flavon channels in  table <ref>.  The reach of the HL-LHC, HE-LHC and a 100 TeV collider is given in table <ref>.  The di-flavon production cross-sections for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries are shown in figure <ref> in the case of soft symmetry-breaking scenario.

The benchmark predictions for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries, are given in tables <ref>- <ref> for  the soft symmetry-breaking scenario and heavy flavon masses.  We observe from tables <ref>- <ref> that the di-flavon production for heavy flavon mass is insensitive to the HL-LHC, HE-LHC and even to a 100 TeV collider for all $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries.

Production cross-sections of the di-flavon in  $\mathcal{Z}_2 \times \mathcal{Z}_{5}$, $\mathcal{Z}_2 \times \mathcal{Z}_{9}$, $\mathcal{Z}_2 \times \mathcal{Z}_{11}$, and $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetries with respect to its mass for 14 TeV HL-LHC, 27 TeV HE-LHC, and 100 TeV hadron collider, where the flavon VEV $f=500$ GeV.

2c|$\mathcal{L}[fb^{-1}]$ [References]     2c|ATLAS 13 TeV     2cCMS 13 TeV

$m_{a}$ [GeV]      cATLAS     c|CMS     c500     c|1000     c500     c1000

$s \rightarrow aa \rightarrow b \bar{b} \ell \ell $ [pb]    $36.1$ [148]      $19.7$ [149]$^{a}$[The reference [149] is at sqrt(s) = 8 TeV.]     $2\e{-2}$     $1\e{-2}$      $2\e{-1}$        $2\e{-2}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]    $36.1$ [150]      $35.9$ [151]      $4\e{-2}$     $9\e{-3}$      $7\e{-2}$       $1\e{-2}$

Current limits of $\sigma \times BR$ at 13 TeV LHC by ATLAS and CMS in high mass resonance searches for di-flavon production channels.

2c|HL-LHC [14 TeV, $3~\iab$]      2c|HE-LHC [27 TeV, $15~\iab$]     2c100 TeV, $30~\iab$

$m_{a}$ [GeV]      500      1000      500      1000      500      1000

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]        $2\e{-3}$     $1\e{-3}$     $2\e{-3}$     $8\e{-4}$     $3\e{-3}$     $1\e{-3}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]      $5\e{-3}$      $1\e{-3}$      $3\e{-3}$     $7\e{-4}$       $5\e{-3}$      $1\e{-3}$

Estimated reach ($\sigma \times BR$) of HL-LHC, HE-LHC and the 100 TeV collider for high flavon mass ($m_a$) in di-flavon production channels.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $1.4\e{-8}$     $4.4\e{-11}$      $7\e{-9}$     $1.7\e{-11}$     $7.5\e{-10}$     $1.7\e{-12}$     $2\e{-9}$     $3.7\e{-12}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $1.3\e{-11}$     $4\e{-14}$      $1.5\e{-10}$     $3.6\e{-13}$     $5.8\e{-12}$     $1.3\e{-14}$     $1.8\e{-11}$     $3.2\e{-14}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $4\e{-8}$      $1.3\e{-10}$     $1.8\e{-8}$     $4.3\e{-11}$     $1.1\e{-9}$     $2.6\e{-12}$      $4.9\e{-9}$     $9.1\e{-12}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 14 TeV HL-LHC, assuming $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $5.9\e{-8}$     $3.7\e{-10}$      $4.2\e{-8}$     $1.9\e{-10}$     $3.4\e{-9}$     $1.5\e{-11}$     $1.6\e{-8}$     $6.9\e{-11}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $5.3\e{-11}$     $3.4\e{-13}$      $9.1\e{-10}$     $4.2\e{-12}$     $2.6\e{-11}$     $1.2\e{-13}$     $1.4\e{-10}$     $5.9\e{-13}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $1.7\e{-7}$      $1.1\e{-9}$     $1.1\e{-7}$     $4.9\e{-10}$     $5.1\e{-9}$     $2.3\e{-11}$      $4\e{-8}$     $1.7\e{-10}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 27 TeV HE-LHC, assuming $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c500     c|1000     c500     c|1000     c500     c|1000    c500     c1000

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $4.8\e{-7}$     $5.5\e{-9}$      $6.5\e{-7}$     $6.2\e{-9}$     $3.3\e{-8}$     $2.8\e{-10}$     $3.7\e{-7}$     $3.5\e{-9}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $4.2\e{-10}$     $4.9\e{-12}$      $1.4\e{-8}$     $1.3\e{-10}$     $2.5\e{-10}$     $2.2\e{-12}$     $3.2\e{-9}$     $3.1\e{-11}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $1.4\e{-6}$      $1.6\e{-8}$     $1.7\e{-6}$     $1.6\e{-8}$     $5\e{-8}$     $4.3\e{-10}$      $9.1\e{-7}$     $8.6\e{-9}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at a 100 TeV collider, assuming $f = 500$ GeV.

We observed earlier that the di-flavon production of a heavy mass flavon is beyond the reach of HL-LHC, HE-LHC, and a 100 TeV collider for different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries.  However, this scenario changes in the case of a light flavon.  We show the present sensitivities of the light di-flavon searches of the LHC in table <ref>.  The sensitivities of the HL-LHC, HE-LHC, and a 100 TeV collider for a light di-flavon productions are shown in table <ref>.

2c|$\mathcal{L}[fb^{-1}]$ [References]     2c|ATLAS 13 TeV     2cCMS 13 TeV

$m_{a}$ [GeV]      cATLAS     c|CMS     c20     c|60     c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \ell \ell $ [pb]    $139$ [152]      $138$ [153]      $1\e{-4}$     $1\e{-4}$      $9\e{-5}$       $8\e{-5}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]    $36.1$ [154]      $138$ [155]      $3$     $1$      $1$       $3\e{-1}$

Current limits of $\sigma \times BR$ at 13 TeV LHC by ATLAS and CMS in low mass resonance searches for di-flavon channels.

2c|HL-LHC [14 TeV, $3~\iab$]      2c|HE-LHC [27 TeV, $15~\iab$]     2c100 TeV, $30~\iab$

$m_{a}$ [GeV]      20      60      20      60      20      60

$s \rightarrow aa \rightarrow b \bar{b} \ell \ell $ [pb]        $2\e{-5}$     $2\e{-5}$     $1\e{-5}$     $1\e{-5}$     $2\e{-5}$     $2\e{-5}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]      0.2      $7\e{-2}$     0.16     $5\e{-2}$       0.2     $7\e{-2}$

Estimated reach ($\sigma \times BR$) of HL-LHC, HE-LHC and the 100 TeV collider for low flavon mass ($m_a$) in di-flavon production channels.

We show our benchmark predictions of  di-flavon production cross-sections  in tables <ref>-<ref> for low flavon masses in the soft symmetry-breaking scenario.  We observe  that $ aa \rightarrow b \bar{b} \tau \tau $ mode is accessible for all $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries at the HL-LHC, HE-LHC, and a 100 TeV collider.  All three channels are sensitive to the HL-LHC, HE-LHC, and a 100 TeV collider only for the $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ flavour symmetry.  The production cross-sections  in the symmetry-conserving scenario are too small to be under the reach of any collider, as observed in tables <ref>-<ref>.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $2.7\e{-5}$     $1.9\e{-5}$      $3.2\e{-5}$     $1.5\e{-5}$     $3.3\e{-5}$     $2\e{-5}$     $1.8\e{-2}$     $1.6\e{-2}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $2.5\e{-8}$     $1.7\e{-8}$      $7.2\e{-7}$     $3.3\e{-7}$     $2.7\e{-7}$     $1.5\e{-7}$     $1.6\e{-4}$     $1.3\e{-4}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $6.8\e{-5}$     $5.4\e{-5}$      $6.1\e{-5}$     $3.7\e{-5}$     $4.1\e{-5}$     $2.9\e{-5}$     $3.6\e{-2}$     $3.7\e{-2}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with low flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 14 TeV HL-LHC, assuming $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $6.8\e{-5}$     $5.5\e{-5}$      $9.5\e{-5}$     $5.8\e{-5}$     $8.9\e{-5}$     $6.1\e{-5}$     $8.8\e{-2}$     $8.1\e{-2}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $6.4\e{-8}$     $5\e{-8}$      $2.2\e{-6}$     $1.3\e{-6}$     $7.1\e{-7}$     $4.7\e{-7}$     $8\e{-4}$     $7\e{-4}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $1.7\e{-4}$     $1.6\e{-4}$      $1.8\e{-4}$     $1.4\e{-4}$     $1.1\e{-4}$     $9.2\e{-5}$     $0.18$     $0.19$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at the 27 TeV HE-LHC, assuming $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     2c$\mathcal{Z}_8 \times \mathcal{Z}_{22}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c|60    c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $3.5\e{-4}$     $3.2\e{-4}$      $6.9\e{-4}$     $5.5\e{-4}$     $5.2\e{-4}$     $4.2\e{-4}$     $1.17$     $1.11$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $3.3\e{-7}$     $2.9\e{-7}$      $1.6\e{-5}$     $1.2\e{-5}$     $4.2\e{-6}$     $3.2\e{-6}$     $1.1\e{-2}$     $9.7\e{-3}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $8.8\e{-4}$     $9.2\e{-4}$      $1.3\e{-3}$     $1.3\e{-3}$     $6.4\e{-4}$     $6.3\e{-4}$     $2.35$     $2.65$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with high flavon mass ($m_a$) in case of the soft symmetry-breaking scenario at a 100 TeV collider, assuming $f = 500$ GeV.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $8.7\e{-10}$     $3.5\e{-12}$      $1.9\e{-14}$     $5.4\e{-17}$     $3.4\e{-16}$     $1.1\e{-18}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $8.2\e{-13}$     $3.2\e{-15}$      $4.2\e{-16}$     $1.2\e{-18}$     $2.7\e{-18}$     $8.8\e{-21}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $2.2\e{-9}$     $1\e{-11}$      $3.6\e{-14}$     $1.3\e{-16}$     $4.2\e{-16}$     $1.7\e{-18}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at the 14 TeV HL-LHC.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $1.9\e{-9}$     $8.8\e{-12}$      $4.7\e{-14}$     $1.6\e{-16}$     $7.7\e{-16}$     $3\e{-18}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $1.8\e{-12}$     $7.9\e{-15}$      $1.1\e{-15}$     $3.6\e{-18}$     $6.2\e{-18}$     $2.3\e{-20}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $4.7\e{-9}$     $2.5\e{-11}$      $9.1\e{-14}$     $4.1\e{-16}$     $9.5\e{-16}$     $3.1\e{-18}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at the 27 TeV HE-LHC.

2c|Benchmark     2c|Benchmark     2cBenchmark

2c|$\mathcal{Z}_2 \times \mathcal{Z}_{5}$     2c|$\mathcal{Z}_2 \times \mathcal{Z}_{9}$     2c$\mathcal{Z}_2 \times \mathcal{Z}_{11}$

$m_{a}$ [GeV]     c20     c|60     c20     c|60     c20     c60

$s \rightarrow aa \rightarrow b \bar{b} \tau \tau $ [pb]          $7.8\e{-9}$     $4.5\e{-11}$      $2.4\e{-13}$     $1.2\e{-15}$     $3.4\e{-15}$     $1.7\e{-17}$

$s \rightarrow aa \rightarrow b \bar{b} \mu \mu $ [pb]          $7.4\e{-12}$     $4\e{-14}$      $5.5\e{-15}$     $2.5\e{-17}$     $2.8\e{-17}$     $1.3\e{-19}$

$s \rightarrow aa \rightarrow b \bar{b}  b \bar{b} $ [pb]         $2\e{-8}$     $1.3\e{-10}$      $4.7\e{-13}$     $2.8\e{-15}$     $4.2\e{-15}$     $2.6\e{-17}$

Benchmark points for different  $\mathcal{Z}_N \times \mathcal{Z}_{M}$  flavour symmetries for di-flavon production channels with low flavon mass ($m_a$) in case of the symmetry-conserving scenario at a 100 TeV collider.

§ SUMMARY

The  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetry  is a unique and novel framework that may have its origin either in a $U(1) \times U(1)$ symmetry or in an anomalous global $U(1)_A \times U(1)_A$ symmetry.  This symmetry provides a unique way to realize the FN mechanism, resulting in an explanation to the flavour problem of the SM.  In this work, we have thoroughly investigated the flavour  phenomenology  and the collider signatures of  different realizations of the  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetry  motivated by different theoretical demands.

In the case of quark flavour physics, the $K^0-\bar K^0$ mixing places the strongest bounds on the parameter space of the  flavon of the $\mathcal{Z}_2 \times \mathcal{Z}_9$ flavour symmetry in the case of the soft symmetry-breaking scenario.  The $\mathcal{Z}_2 \times \mathcal{Z}_5$ flavour symmetry  is subjected to the weakest constraints from the  $K^0-\bar K^0$ mixing.  This scenario continues to remain the same even for the symmetry-conserving case, where the allowed parameter space is only along a straight dashed line.  The  $B_s -\bar B_s$ mixing better constraints the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$  and $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetries for the soft symmetry-breaking scenario. In the case of the symmetry-conserving potential, the $B_s -\bar B_s$ mixing provides more stringent bounds on the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ flavour symmetry.  The  $B_d -\bar B_d$ mixing places more tight bounds on the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$  and $\mathcal{Z}_{2} \times \mathcal{Z}_{5,9}$ flavour symmetries for the soft symmetry-breaking scenario.  This feature continues to hold even for the symmetry-conserving potential. The   $D^0 - \bar D^0$ mixing  works remarkably well in constraining the  parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{5}$ flavour symmetry for the soft symmetry-breaking scenario.  This is also true for the symmetry-conserving case.  The $B_{d,s} \rightarrow \mu^+ \mu^-$ decays impose relatively better bounds on the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ symmetry.  On the other side, the $ K_L \rightarrow \mu^+ \mu^-$ decays are more sensitive to the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{9}$ symmetry.   The $B_{d,s} \rightarrow \mu^+ \mu^-$  and $ K_L \rightarrow \mu^+ \mu^-$ decays  place tighter constraints on the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ symmetry for the symmetry-conserving scenario.  The lifetime $\tau_{\mu\mu}$ is important in constraining the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{5}$ symmetry.  For the soft symmetry-breaking scenario, the lifetime $\tau_{\mu\mu}$ places more significant bounds on the parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ symmetry.  The ratio $R_{\mu \mu}$ is a powerful observable to constrain the  parameter space of the  flavon of all  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries for the soft symmetry-breaking as well as symmetry-conserving scenarios, and will play a crucial role in determining the parameter space of the flavon of different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries.  A global fit to diverse data coming from the observables having $b \rightarrow s \mu^+ \mu^-$ transitions is performed, and allowed ranges of the Wilson coefficients are derived.  This results in  bounds on the parameter space of the  flavon of all  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries for the soft symmetry-breaking as well as symmetry-conserving scenarios. We also predict muon forward-backward asymmetry in $B \rightarrow K \mu^+ \mu^-$ decays for $f<200$ GeV.

On the leptonic flavour side, the decay $\mu \rightarrow e \gamma$ is able to constrain more the  parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{9}$  and the  $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetries for  the soft symmetry-breaking case.  However, the  parameter space of the  flavon of the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ is more constrained for the symmetry-conserving scenario.  The $A~\mu\rightarrow A~e$ conversion is  more effective in constraining the  parameter space of the  flavon of the $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetry for the soft symmetry-breaking scenario.  In the case of symmetry-conserving scenario, the parameter space of the  flavon of the  $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ is more constrained.  Similar results are obtained for the $\mu\rightarrow 3e$ and $\tau \rightarrow 3 \mu$ decays.

In the study of collider physics of the flavon of different $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries, we computed flavon decays as well as $t \rightarrow (c,u) a $ decays.  The  $t \rightarrow c a $ decays are within the reach of the HL-LHC, HE-LHC, and a 100 TeV collider only for the $\mathcal{Z}_{2} \times \mathcal{Z}_{11}$ and  the $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetries for the soft symmetry-breaking scenario. Furthermore,  we  have investigated inclusive, associative, and di-flavon production signatures of flavon  for the soft symmetry-breaking as well as symmetry-conserving scenarios.  We have predicted the sensitivities  of these channels for the HL-LHC, HE-LHC, and a 100 TeV collider in a model-independent way using the square root scaling of luminosity.   We compare our benchmark predictions for heavy and light flavon at the HL-LHC, HE-LHC, and a 100 TeV collider to estimated sensitivities of  the HL-LHC, HE-LHC and a 100 TeV collider for all the  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries discussed in this work.  In the case of inclusive production channels, a 100 TeV collider can probe the $\mathcal{Z}_{2} \times \mathcal{Z}_{5,9}$ and  the $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetries for the soft symmetry-breaking scenario for a heavy flavon. The HE-LHC will be able to probe the $\mathcal{Z}_{2} \times \mathcal{Z}_{5}$ and  the $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour symmetries better than the HL-LHC for the soft symmetry-breaking scenario. In the case of a light flavon, the HE-LHC and a 100 TeV collider can probe all the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries in the $\tau \tau$ channel for the soft symmetry-breaking scenario. A few specific  inclusive signatures are within the reach of the 14 TeV high-luminosity LHC  only for the $\mathcal{Z}_{\rm 2} \times \mathcal{Z}_{\rm 5}$ and $\mathcal{Z}_{\rm 8} \times \mathcal{Z}_{\rm 22}$  flavour symmetries.  The symmetry-conserving scenario is beyond the reach of detection capabilities of any collider for the inclusive processes.

For associative production modes, only the  parameter space of the  flavon of the $\mathcal{Z}_{8} \times \mathcal{Z}_{22}$ flavour is accessible to  the HL-LHC, HE-LHC, and a 100 TeV collider  through the  $t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $  channel in the soft symmetry-breaking scenario for  high flavon mass. The di-flavon production channels for heavy flavon are insensitive to  the HL-LHC, HE-LHC, and even to a 100 TeV collider for all the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries in the case of soft symmetry-breaking scenario.  However, a light flavon scenario can be probed for all the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetries at the HL-LHC, HE-LHC, and a 100 TeV collider. In the case of the symmetry-conserving scenario, the di-flavon productions cross-sections are too small to be accessible at the HL-LHC, HE-LHC as well as a 100 TeV collider, and can be further constrained through the precision flavour physics in future.  Collider simulations for some particular signatures, such as $pp \rightarrow a \rightarrow t \bar{t}$,   could be an interesting future investigation, which is an expected sequel to the present work [156].

§ ACKNOWLEDGEMENT

We acknowledge the use of JaxoDraw [157] for creating the Feynman diagrams in this work.  This work is supported by the  Council of Science and Technology,  Govt. of Uttar Pradesh,  India through the  project “   A new paradigm for flavour problem "  no.   CST/D-1301,  and Science and Engineering Research Board,  Department of Science and Technology, Government of India through the project “ Higgs Physics within and beyond the Standard Model" no. CRG/2022/003237. NS acknowledges the support through the INSPIRE fellowship by the Department of Science and Technology, Government of India.

§ BENCHMARK POINTS FOR THE YUKAWA COUPLINGS

We reproduce the fermion masses using the following values of the fermion masses at $ 1$TeV[158],

\begin{eqnarray}
\{m_t, m_c, m_u\} &\simeq& \{150.7 \pm 3.4,~ 0.532^{+0.074}_{-0.073},~ (1.10^{+0.43}_{-0.37}) \times 10^{-3}\}~{\rm GeV}, \nonumber \\
\{m_b, m_s, m_d\} &\simeq& \{2.43\pm 0.08,~ 4.7^{+1.4}_{-1.3} \times 10^{-2},~ 2.50^{+1.08}_{-1.03} \times 10^{-3}\}~{\rm GeV},
\nonumber \\
\{m_\tau, m_\mu, m_e\} &\simeq& \{1.78\pm 0.2,~ 0.105^{+9.4 \times 10^{-9}}_{-9.3 \times 10^{-9}},~ 4.96\pm 0.00000043 \times 10^{-4}\}~{\rm GeV}.
\end{eqnarray}

The magnitudes and phases  of the CKM mixing elements are [50],
|V_ud|    =    0.97370 ±0.00014,  |V_cb| = 0.0410 ±0.0014, |V_ub| = 0.00382 ±0.00024,
sin2 β   =    0.699 ±0.017,   α= (84.9^+5.1_-4.5)^∘,   γ= (72.1^+4.1_-4.5)^∘, δ= 1.196^+0.045_-0.043

The dimensionless coefficients $y_{ij}^{u,d,\ell,\nu}= |y_{ij}^{u,d,\ell,\nu}| e^{i \phi_{ij}^{q,\ell,\nu}}$  are scanned with $|y_{ij}^{u,d,\ell, \nu}| \in [0.7, 2 \pi]$ and $ \phi_{ij}^{q,\ell,\nu} \in [0,2\pi]$. The fit results  are,

§.§ The $\mathcal{Z}_2 \times \mathcal{Z}_{5}$ model

The dimensionless coefficients $y_{ij}^{u,d,\ell,\nu}= |y_{ij}^{u,d,\ell,\nu}| e^{i \phi_{ij}^{q,\ell,\nu}}$  are scanned with $|y_{ij}^{u,d,\ell, \nu}| \in [0.1, 4 \pi]$ and $ \phi_{ij}^{q,\ell,\nu} \in [0,2\pi]$. The fit results  are,

\begin{equation*}
Y_u =\begin{pmatrix}
-1.68 - 3.37 i &  -0.09 + 0.03 i  &   -0.1 - 0.02 i    \\
1.53 + 4.95 i    &- 0.57 + 0.55 i  &  0.48 + 0.002 i    \\
0.76 + 0.18 i   &  -1.04 + 0.46 i   &  0.58 - 0.65 i
\end{pmatrix},
Y_d = \begin{pmatrix}
-4.15 + 3.58 i &  2.20 - 0.89 i & 2.62 - 4.20 i   \\
-0.33 - 0.36 i  & 0.07 -0.075 i &  0.17 + 0.47 i  \\
- 0.24 -0.07 i &  -0.06 - 0.084 i   &  -0.07 - 0.12 i
\end{pmatrix},  \\
\end{equation*}

\begin{equation*}
Y_l = \begin{pmatrix}
-0.07 - 0.06 i &  0.099 - 0.004 i & 0.45 -0.32   i   \\
-0.14 - 0.09 i  & 0.08 - 0.06 i &  -0.63 + 0.24 i  \\
-0.04 + 0.09 i &  -0.09 + 0.06 i   &  0.10 - 0.0003 i
\end{pmatrix}
\end{equation*}

with $\epsilon = 0.1$ and $\delta = 1.196$.

§.§ The $\mathcal{Z}_2 \times \mathcal{Z}_{9}$ model

The dimensionless coefficients $y_{ij}^{u,d,\ell,\nu}= |y_{ij}^{u,d,\ell,\nu}| e^{i \phi_{ij}^{q,\ell,\nu}}$  are scanned with $|y_{ij}^{u,d,\ell, \nu}| \in [0.9, 2 \pi]$ and $ \phi_{ij}^{q,\ell,\nu} \in [0,2\pi]$. The fit results  are,

\begin{equation*}
Y_u =\begin{pmatrix}
1  &  0.87 - 0.49 i  &   -0.23 + 0.97 i    \\
-0.9 + 1.05 i    &- 0.7 - 0.72 i  &  1    \\
0.94 - 0.33 i   &  0.55 + 0.84 i   &  0.9
\end{pmatrix},
Y_d = \begin{pmatrix}
0.99 - 0.09 i &  3.24 - 1.05 i & 1    \\
0.99 - 0.10 i  & 0.92 + 0.39 i &  0.9   \\
1  &  1    &  -1.04 + 0.54 i
\end{pmatrix},  \\
\end{equation*}

\begin{equation*}
Y_l = \begin{pmatrix}
0.9  &  0.9  & 1.5    \\
0.9   & 1.5  &  1.5  \\
1.5  &  1.5    &  0.9
\end{pmatrix}
\end{equation*}

with $\epsilon = 0.23$ and $\delta = 1.196$.

§.§ The $\mathcal{Z}_2 \times \mathcal{Z}_{11}$ model

The dimensionless coefficients $y_{ij}^{u,d,\ell,\nu}= |y_{ij}^{u,d,\ell,\nu}| e^{i \phi_{ij}^{q,\ell,\nu}}$  are scanned with $|y_{ij}^{u,d,\ell, \nu}| \in [0.7, 2 \pi]$ and $ \phi_{ij}^{q,\ell,\nu} \in [0,2\pi]$. The fit results  are,

\begin{equation*}
Y_u =\begin{pmatrix}
2.19 - 0.18 i &  1.40 - 0.09 i  &   0.70 - 0.002 i    \\
2.12 + 0.02 i    &- 1.02 + 2.12 i  &  4.45 - 1.01 i    \\
0.33 - 0.71 i   &  0.92 + 0.42 i   &  0.82 - 0.26 i
\end{pmatrix},
Y_d = \begin{pmatrix}
0.40 + 0.59 i &  3.61 - 0.16 i & 3.34 - 4.06 i   \\
1.02 + 0.01 i  & -0.36 + 1.18 i &  1.29 + 0.36 i  \\
- 0.77 + 0.33 i &  0.82 + 0.02 i   &  -0.51 + 0.48 i
\end{pmatrix}  \\
\end{equation*}

\begin{equation*}
Y_l = \begin{pmatrix}
-0.73 + 0.001 i &  -0.58 + 0.53 i & 0.001 + 1.29  i   \\
6.24 + 0.37 i  & 3.48 - 2.44 i &  1.71 + 1.7 \times 10^{-5} i  \\
0.24 - 1.16 i &  -0.36 - 0.61 i   &  -0.60 + 0.35 i
\end{pmatrix}  \\
\end{equation*}

with $\epsilon = 0.28$ and $\delta = 1.196$.

§.§ The $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ model

The dimensionless coefficients $y_{ij}^{u,d,\ell,\nu}= |y_{ij}^{u,d,\ell,\nu}| e^{i \phi_{ij}^{q,\ell,\nu}}$  are scanned with $|y_{ij}^{u,d,\ell, \nu}| \in [0.9, 2]$ and $ \phi_{ij}^{q,\ell,\nu} \in [0,2\pi]$. The fit results  are,

\begin{equation*}
Y_u =\begin{pmatrix}
0.12 + 1.44 i &  - 0.38 - 0.82 i  &   0.989 + 0.004 i    \\
- 1.27 - 1.38 i    & - 0.56 + 0.82 i &  - 1.22 - 0.24 i     \\
- 1.12 - 0.25 i    &  - 1.12 - 0.43 i   &  - 2.70 - 2.61 i
\end{pmatrix},
Y_d = \begin{pmatrix}
- 1.36 + 0.39 i &  0.31 - 1.28 i & 0.71 + 0.61 i   \\
- 1.05 + 0.27 i  & -0.44 + 0.85 i &  0.47 - 0.79 i  \\
- 1.11 - 0.37 i & - 0.81 + 0.42 i   &  0.80 + 0.82 i
\end{pmatrix}  \\
\end{equation*}

\begin{equation*}
Y_l = \begin{pmatrix}
-1.40 - 0.21 i &  1.14 - 0.004 i & - 0.78 + 0.45  i   \\
- 0.88 + 0.16 i  & - 0.54 + 0.78 i &  - 1.16 + 0.11 i  \\
0.85 + 0.34 i &  0.899 + 0.003 i   &  0.899 - 0.003 i
\end{pmatrix}  \\
\end{equation*}

with $\epsilon = 0.23$ and $\delta = 1.196$.

§ THE SM BACKGROUND

Production mode     Channel      SM Backgrounds
jet-jet     $p p \rightarrow j j$

$\ell^+ \ell^- (\ell = e, \mu, \tau)$     $p p \rightarrow \ell^+ \ell^-$

$\mu e $     $p p \rightarrow t \bar{t}, p p \rightarrow V V (V= W^+, W^-, Z),$
$p p \rightarrow \ell^+ \ell^- , p p \rightarrow t W b$

Inclusive      $\mu \tau, e \tau $     $p p \rightarrow$ multijets,  $p p \rightarrow W +$ jets ,

production         $p p \rightarrow t \bar{t}, p p \rightarrow V V (V= W^+, W^-, Z),$         $p p \rightarrow \ell^+ \ell^- , p p \rightarrow t W b$

$b \overline{b}$     $p p \rightarrow b \bar{b}$

$\gamma \gamma$     $p p \rightarrow \gamma \gamma$

$t \overline{t}$     $p p \rightarrow t \bar{t}$

Associative     $t  \bar{t} a \rightarrow t  \bar{t} t  \bar{t} $     $p p \rightarrow t  \bar{t} t  \bar{t} $

production     $a b \bar{b} \rightarrow \tau \tau b \bar{b}$     $p p \rightarrow \tau \tau b \bar{b}$

$a b  \rightarrow \tau \tau b $      $p p \rightarrow$ multijets, $p p \rightarrow W + $ jets, $p p \rightarrow t \bar{t} $
$p p \rightarrow \ell^+ \ell^- , p p \rightarrow t W b$, $p p \rightarrow V V$

Di-flavon     $a  a \rightarrow b  \bar{b} \ell  \ell (\ell = \mu, \tau) $     $p p \rightarrow  b  \bar{b} \ell  \ell $

production     $a a \rightarrow b \bar{b} b \bar{b}$     $p p \rightarrow b \bar{b} b \bar{b}$

SM Backgrounds for the flavon production through various channels.

§ THE DARK-TECHNICOLOR PARADIGM

In this appendix, we show how the  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetry and the FN mechanism of the  $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ model naturally emerge from a dark-technicolour paradigm [3, 4, 6]. The technicolour paradigm is obtained by $SU(\rm N_{\rm TC}) \times SU(\rm N_{\rm DTC}) \times SU(\rm{N}_{\rm F})$, where TC denotes the standard  technicolour gauge group,  DTC is for the dark-technicolour, and the F stands for the  strong dynamics of vector-like fermions.

The TC fermions  transform under the  symmetry $\rm SM \times \mathcal{G}$  as[4],

\begin{eqnarray}
T_{q}^i  &\equiv&   \begin{pmatrix}
T  \\
\end{pmatrix}_L:(1,2,0,\rm{N}_{\rm TC},1,1),  \\ \nonumber
T_{R}^i &:& (1,1,1,\text{N}_{\rm{TC}},1,1), B_{R}^i : (1,1,-1,\rm{N}_{\rm TC},1,1),
\end{eqnarray}

where $i=1,2,3 \cdots $,  and   $+\frac{1}{2}$ is the electric charge for $T$, and $-\frac{1}{2}$ is that of the  $B$.

The transformation of the DTC fermions under the  $\rm SM \times \mathcal{G}$  symmetry is [4],

\begin{eqnarray}
\mathcal{D}_{ q}^i &\equiv& \mathcal{C}_{L,R}^i  : (1,1, 1,1,\text{N}_{\text{DTC}},1),~\mathcal{S}_{L,R}^i  : (1,1,-1,1,\text{N}_{\text{DTC}},1),
\end{eqnarray}

where  $i=1,2,3 \cdots $,  and  electric charges are $+\frac{1}{2}$ for $\mathcal C$ and $-\frac{1}{2}$ for $\mathcal S$.

The fermions of the $SU(N_{\text{F}})$ symmetry have the following behavior under the $\rm SM \times \mathcal{G}$  symmetry [4],

\begin{eqnarray}
F_{L,R} &\equiv &U_{L,R}^i \equiv  (3,1,\dfrac{4}{3},1,1,\text{N}_\text{F}),
D_{L,R}^{i} \equiv   (3,1,-\dfrac{2}{3},1,1,\text{N}_\text{F}),  \\ \nonumber
N_{L,R}^i &\equiv&   (1,1,0,1,1,\text{N}_\text{F}),
E_{L,R}^{i} \equiv   (1,1,-2,1,1,\text{N}_\text{F}),
\end{eqnarray}

where  $i=1,2,3 \cdots$.

We notice that there are  three axial $U(1)_A^{\rm TC, DTC,  F}$ symmetries in the dark-technicolor paradigm.  They are broken by instanton effects to a cyclic discrete group as $ U(1)_A^{\rm TC, DTC, F} \rightarrow \mathcal{Z}_{2 \rm K_{\rm TC, DTC, F}}$ [159],  where $\rm K_{\rm TC, DTC, F} $ show the number of   massless flavours of the TC, DTC and F gauge symmetries  in the $N$-dimensional representation of the gauge group $SU(\rm N)_{\rm TC, DTC, F}$.  Thus, we observe a natural emergence of a generic  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} \times \mathcal{Z}_{\rm P}$ flavour symmetry, whose subset is the  $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M} $ flavour symmetry.

In the next step, we assume the existence of an extended technicolour  dynamics (ETC) which accommodates the SM, TC, and DTC sectors. This provides the required interactions for the masses of fermions in the FN mechanism.  These interactions are shown in figure <ref>, where $\langle \varphi \rangle$ and $\langle \chi \rangle$  show the Higgs and flavon VEVs, respectively,  in the form of an on-shell chiral condensate.

Feynman diagrams for the masses of the quarks and charged leptons in the  dark-technicolour paradigm.  On the top, the generic interactions of the SM, TC, F and DTC fermions are shown.  In the bottom, we see the formations of the different condensates (circular blobs),    and the resulting mass of the SM fermions.

The multi-fermion condensate can be written as  [160],

⟨( ψ̅_L ψ_R )^n ⟩∼(  Λexp(k Δχ) )^3n,
where $\Delta \chi$ is the chirality of an operator, $k$  is a constant, and $\Lambda$ represents the scale of the underlying gauge theory.

Now, the masses of the SM fermions in the FN mechanism can be written as,

m_f     =       Λ_TC^3/Λ_ETC^2 ( 1Λ Λ_DTC^3/Λ_ETC^2 exp(2 k) )^n_ij^f,
where $f=u,d$, and $\Lambda_{\text{TC}}$, $\Lambda_{\text{DTC}}$ and  $\Lambda_F $ are the scale of the TC, DTC, and F dynamics, respectively.  We can now identify the order parameter $\epsilon$ as,

\begin{align}
\epsilon = \frac{\langle \chi \rangle}{\Lambda_F}   = \dfrac{1}{\Lambda_F} \frac{\Lambda_{\text{DTC}}^3}{\Lambda_{\text{ETC}}^{2}} \exp(2 k),
\end{align}

and the SM Higgs VEV is given by,

\begin{align}
\langle \varphi \rangle =   \frac{\Lambda_{\text{TC}}^{3}}{\Lambda_{\text{ETC}}^2}.
\end{align}

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Observables                        $\mathcal{Z}_2 \times \mathcal{Z}_{9}$     $\mathcal{Z}_2 \times \mathcal{Z}_{11}$      $\mathcal{Z}_8 \times \mathcal{Z}_{12}$

$C_{\Delta m_K}$

$C_{B_{s}}$ (current)

$C_{B_{s}}$ (Phase- 1)

$C_{B_{s}}$ (Phase- 2)

$C_{B_{d}}$ (current)

$C_{B_{d}}$ (Phase- 1)

$C_{B_{d}}$ (Phase- 2)

$\br(B_s\rightarrow \mu^+\mu^-) $

$ \mathcal{R}_{\mu\mu}$

$\br(B_d\rightarrow \mu^+\mu^-) $
$\br(K_L\rightarrow \mu^+\mu^-) $
$\br(D\rightarrow \mu^+\mu^-) $
$\br(\mu \rightarrow e \gamma)$ (Current)

$\br(\mu \rightarrow e \gamma)$ (Projected)
$\br(\mu \rightarrow 3 e )$ (Current)

$\br(\mu \rightarrow 3 e)$ (Projected)

$\mathcal{Z}_N \times \mathcal{Z}_M$ symmetry      Constraining flavour observables
$\mathcal{Z}_2 \times \mathcal{Z}_5$     BR $  \text{(} \mu  \to e  \text{)}^{\rm Ti}_{(10^{-20})}$  $<$ $\Phi_{B_s}$ $<$ $\tau_{\mu \mu}$ $<$ $\Phi_{B_d}$ $<$ $C_{B_d}^{\rm{LHCb-II}}$ $<$ $C_{\eps_K}$ $<$ $|M_{12}^D|$  $<$ $R_{\mu \mu}^{\rm{LHCb-II}}$

$\mathcal{Z}_2 \times \mathcal{Z}_9$     $\tau_{\mu \mu}$  $<$ $|M_{12}^D|$ $<$ $C_{B_s}^{\rm{LHCb-II}}$ $<$ $\Phi_{B_d}$ $<$ BR $  \text{(} \mu  \to e  \text{)}^{\rm Ti}_{(10^{-20})}$ $<$ $C_{B_d}^{\rm{LHCb-II}}$ $<$ $C_{\eps_K}$ $<$ $C_{\Delta m_K}^{\rm{LHCb-I}}$ $<$ $R_{\mu \mu}^{\rm{LHCb-II}}$

$\mathcal{Z}_2 \times \mathcal{Z}_{11}$       $\tau_{\mu \mu}$  $<$ $|M_{12}^D|$ $\approx$ $C_{B_s}^{\rm{LHCb-II}}$ $<$ BR $  \text{(} \mu  \to e  \text{)}^{\rm Ti}_{(10^{-20})}$ $<$ $C_{\Delta m_K}^{\rm{LHCb-I}}$ $<$ $C_{B_d}^{\rm{LHCb-II}}$ $\approx$ $C_{\eps_K}$  $<$ $R_{\mu \mu}^{\rm{LHCb-II}}$

$\mathcal{Z}_8 \times \mathcal{Z}_{22}$     $\tau_{\mu \mu}$ $<$ $\Phi_{B_d}$ $\approx$ $\Phi_{B_s}$ $<$ $C_{B_s}^{\rm{LHCb-II}}$ $\approx$ $C_{B_d}^{\rm{LHCb-II}}$ $<$ $|M_{12}^D|$ $\approx$ $C_{\Delta m_K}^{\rm{LHCb-I}}$ $<$ BR $  \text{(} \mu  \to e  \text{)}^{\rm Ti}_{(10^{-20})}$  $<$ $C_{\eps_K}$  $<$ $R_{\mu \mu}^{\rm{LHCb-II}}$

$\mathcal{Z}_N \times \mathcal{Z}_M$ symmetry      flavour observable     2c|Benchmark values

$m_a$ $({\rm GeV})$     $f$ $({\rm GeV})$

$\mathcal{Z}_2 \times \mathcal{Z}_5$     $C_{\Delta m_K}$      100     100

$\mathcal{Z}_2 \times \mathcal{Z}_9$     $|M_{12}^D|$     100     100

$\mathcal{Z}_2 \times \mathcal{Z}_{11}$     $C_{\eps_K}$     100     100

$\mathcal{Z}_8 \times \mathcal{Z}_{22}$     $C_{B_d}$     100     100

The couplings of the pseudoscalar  component field $a$  with fermions for the minimal $\mathcal{Z}_2 \times \mathcal{Z}_5$ symmetry are written as,

\begin{eqnarray}
\label{fup}
y_{af_{iL} f_{jR}}^{u} &\equiv & y_{aij}^u = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
4 y_{11}^u  \epsilon^4 & 4 y_{12}^u \epsilon^4  &  4 y_{13}^u \epsilon^4    \\
2 y_{21}^u  \epsilon^2    & 2 y_{22}^u \epsilon^2  &  2 y_{23}^u \epsilon^2     \\
0   & 0   & 0
\end{pmatrix},
y_{aij}^{d} = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
5 y_{11}^d  \epsilon^5 & 5 y_{12}^d \epsilon^5 & 5 y_{13}^d \epsilon^5   \\
3 y_{21}^d  \epsilon^3  & 3 y_{22}^d \epsilon^3 & 3 y_{23}^d \epsilon^3  \\
y_{31}^d \epsilon &   y_{32}^d \epsilon   &  y_{33}^d \epsilon
\end{pmatrix}, \\ \nonumber
y_{aij}^{\ell} &=& \frac{v}{\sqrt{2}f}
\begin{pmatrix}
5 y_{11}^\ell  \epsilon^5 & 5 y_{12}^\ell \epsilon^5 & 5 y_{13}^\ell \epsilon^5   \\
3 y_{21}^\ell  \epsilon^3  & 3 y_{22}^\ell \epsilon^3 & 3 y_{23}^\ell \epsilon^3  \\
y_{31}^\ell \epsilon &   y_{32}^\ell \epsilon   &  y_{33}^\ell \epsilon
\end{pmatrix}.
\end{eqnarray}

The couplings matrices in the non-minimal $\mathcal{Z}_2 \times \mathcal{Z}_9$ symmetry are,

\begin{eqnarray}
\label{fup_z2z9}
y_{af_{iL} f_{jR}}^{u} &\equiv & y_{aij}^u = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
8 y_{11}^u  \epsilon^8 & 6 y_{12}^u \epsilon^6  &  8 y_{13}^u \epsilon^8    \\
8 y_{21}^u  \epsilon^8    & 4 y_{22}^u \epsilon^4  &  8 y_{23}^u \epsilon^8  \\
0   & 4 y_{32}^u \epsilon^4   & 0
\end{pmatrix},
y_{aij}^{d} = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
7 y_{11}^d  \epsilon^7 & 7 y_{12}^d \epsilon^7 & 7 y_{13}^d \epsilon^7   \\
5 y_{21}^d  \epsilon^5  & 5 y_{22}^d \epsilon^5 & 5 y_{23}^d \epsilon^5  \\
3 y_{31}^d \epsilon^3 &  3 y_{32}^d \epsilon^3   &  3 y_{33}^d \epsilon^3
\end{pmatrix}, \\ \nonumber
y_{aij}^{\ell} &=& \frac{v}{\sqrt{2}f}
\begin{pmatrix}
7 y_{11}^\ell  \epsilon^7 & 7 y_{12}^\ell \epsilon^7 & 7 y_{13}^\ell \epsilon^7   \\
5 y_{21}^\ell  \epsilon^5  & 5 y_{22}^\ell \epsilon^5 & 5 y_{23}^\ell \epsilon^5  \\
3 y_{31}^\ell \epsilon^3 &  3 y_{32}^\ell \epsilon^3   & 3 y_{33}^\ell \epsilon^3
\end{pmatrix}.
\end{eqnarray}

For the $\mathcal{Z}_2 \times \mathcal{Z}_{11}$ symmetry, the couplings are given by,

\begin{eqnarray}
\label{fup_z2z11}
y_{af_{iL} f_{jR}}^{u} &\equiv & y_{aij}^u = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
10 y_{11}^u  \epsilon^{10} & 10 y_{12}^u \epsilon^{10}  &  12 y_{13}^u \epsilon^{12}    \\
6 y_{21}^u  \epsilon^6    & 6 y_{22}^u \epsilon^6  &  4 y_{23}^u \epsilon^4  \\
2 y_{31}^u \epsilon^2  & 2 y_{32}^u \epsilon^2  & 0
\end{pmatrix},
y_{aij}^{d} = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
9 y_{11}^d  \epsilon^9 & 9 y_{12}^d \epsilon^9 & 9 y_{13}^d \epsilon^9   \\
7 y_{21}^d  \epsilon^7  & 7 y_{22}^d \epsilon^7 & 7 y_{23}^d \epsilon^7  \\
3 y_{31}^d \epsilon^3 &  3 y_{32}^d \epsilon^3   &  3 y_{33}^d \epsilon^3
\end{pmatrix},  \\ \nonumber
y_{aij}^{\ell} &=& \frac{v}{\sqrt{2}f}
\begin{pmatrix}
9 y_{11}^\ell  \epsilon^9 & 9 y_{12}^\ell \epsilon^9 & 9 y_{13}^\ell \epsilon^9   \\
7 y_{21}^\ell  \epsilon^7  & 7 y_{22}^\ell \epsilon^7 & 7 y_{23}^\ell \epsilon^7  \\
3 y_{31}^\ell \epsilon^3 &  3 y_{32}^\ell \epsilon^3   & 3 y_{33}^\ell \epsilon^3
\end{pmatrix}.
\end{eqnarray}

For $\mathcal{Z}_8 \times \mathcal{Z}_{22}$ symmetry, the couplings matrices are,

\begin{eqnarray}
\label{fup_z8z22}
y_{af_{iL} f_{jR}}^{u} &\equiv & y_{aij}^u = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
8 y_{11}^u  \epsilon^{8} & 5 y_{12}^u \epsilon^{5}  &  4 y_{13}^u \epsilon^{4}    \\
7 y_{21}^u  \epsilon^7    & 4 y_{22}^u \epsilon^4  &  3 y_{23}^u \epsilon^3  \\
5 y_{31}^u \epsilon^5  & 2 y_{32}^u \epsilon^2  & y_{33}^u \epsilon
\end{pmatrix},
y_{aij}^{d} = \frac{v}{\sqrt{2}f}
\begin{pmatrix}
7 y_{11}^d  \epsilon^7 & 6 y_{12}^d \epsilon^6 & 6 y_{13}^d \epsilon^6   \\
6 y_{21}^d  \epsilon^6  & 5 y_{22}^d \epsilon^5 & 5 y_{23}^d \epsilon^5  \\
4 y_{31}^d \epsilon^4 &  3 y_{32}^d \epsilon^3   &  3 y_{33}^d \epsilon^3
\end{pmatrix}, \\ \nonumber
y_{aij}^{\ell} &=& \frac{v}{\sqrt{2}f}
\begin{pmatrix}
9 y_{11}^\ell  \epsilon^9 & 6 y_{12}^\ell \epsilon^6 & 4 y_{13}^\ell \epsilon^4   \\
8 y_{21}^\ell  \epsilon^8  & 5 y_{22}^\ell \epsilon^5 & 3 y_{23}^\ell \epsilon^3  \\
8 y_{31}^\ell \epsilon^8 &  5 y_{32}^\ell \epsilon^5   & 3 y_{33}^\ell \epsilon^3
\end{pmatrix}.
\end{eqnarray}