# Phenomenology at the LHC of composite bosons from strongly interacting
Standard Model fermions via four-fermion operators of NJL type

S. Ajmal Dipartimento di Fisica e Geologia, Univeristà degli Studi di Perugia,
Italy  INFN, Sezione di Perugia, Via A. Pascoli, I-06123, Perugia, Italy  J.
T. Gaglione Department of Physics and Astronomy, Vanderbilt University,
Nashville, TN, 37235, USA  A. Gurrola Department of Physics and Astronomy,
Vanderbilt University, Nashville, TN, 37235, USA  R. Leonardi INFN, Sezione di
Perugia, Via A. Pascoli, I-06123, Perugia, Italy  O. Panella INFN, Sezione di
Perugia, Via A. Pascoli, I-06123, Perugia, Italy  M. Presilla INFN, Sezione di
Perugia, Via A. Pascoli, I-06123, Perugia, Italy  F. Romeo Department of
Physics and Astronomy, Vanderbilt University, Nashville, TN, 37235, USA  H.
Sun Institute of Theoretical Physics, School of Physics, Dalian University of
Technology, No.2 Linggong Road, Dalian, Liaoning, 116024, P.R.China  S. S. Xue
<EMAIL_ADDRESS>(Corresponding author) ICRANet, Piazzale della Repubblica,
10-65122, Pescara, Italy  Physics Department, Sapienza University of Rome,
Piazzale Aldo Moro 5, 00185 Roma, Italy  INFN, Sezione di Perugia, Via A.
Pascoli, I-06123, Perugia, Italy

###### Abstract

abstract………………………….

###### pacs:

12.60.-i,12.60.Rc,14.80.-j

###### Contents

  1. I Introduction
  2. II LQ from four-fermion operators and contact interactions
  3. III Signal implementation and generation
  4. IV Phenomenology of LQ at the LHC
  5. V Sensitivity at LHC and HL-LHC
  6. VI Summary and remarks
  7. VII Four-fermion operators and contact interactions
    1. VII.1 Composite bosons $\Pi^{0,\pm}$
    2. VII.2 Quark-lepton composite bosons $\Pi^{5/3}_{a}$, $\Pi^{-1/3}_{a}$ and $\Pi^{2/3}_{u_{a}}$, $\Pi^{2/3}_{d_{a}}$
    3. VII.3 Contact interaction of composite bosons, quarks and leptons
  8. VIII LQ Composite fermions $F_{a}$
    1. VIII.1 The type 1 of $G\left[(\bar{\ell}^{i}_{L}e_{R})(\bar{d}^{a}_{R}\psi_{Lia})+(\bar{\ell}^{i}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}\psi_{Lia})\right]+{\rm h.c.}$ (146)
    2. VIII.2 The type 3 of $G\left[(\bar{\psi}_{Lia}\nu^{e}_{R})(\bar{\nu}^{e}_{R}\psi_{Lia})+(\bar{\psi}_{Lia}e_{R})(\bar{e}_{R}\psi_{Lia})\right]+{\rm h.c.}$ (4)
    3. VIII.3 The type 2 of $G\left[(\bar{\ell}^{i}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\ell_{Li})+(\bar{\ell}^{i}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\ell_{Li})\right]+{\rm h.c.}$ (3)
  9. IX contact interaction of composite fermions
    1. IX.1 Contact interaction of $\Pi^{0,\pm}$ composite boson and gauge bosons
    2. IX.2 Contact interaction of Higgs-like ${\rm H}^{0,\pm}$ composite boson and gauge bosons
  10. X Phenomenology of the composite bosons in $pp$ and $ep$ collisions
    1. X.1 Production and decay of composite bosons $\Pi^{0,\pm}$
    2. X.2 Production and decay of quark-lepton composite bosons $\Pi^{5/3,-1/3,2/3}_{a}$
    3. X.3 Production and decay of quark-lepton composite bosons $\Pi^{5/3,-1/3,2/3}_{a}$ in $pp$ collisions
    4. X.4 Vector-boson scatterings in LHC
    5. X.5 Cross sections, decay widths and branching ratios
      1. X.5.1 Cross sections
      2. X.5.2 Decay widths
      3. X.5.3 Branching ratios
    6. X.6 Parameters are $m_{{}_{\Pi}}$ and $F_{\Pi}$
    7. X.7 Wide jets and photons from composite bosons $\Pi^{0}_{u,d}$ production and decay
  11. XI Search for $\Pi$ at the LHC
    1. XI.1 Branching ratios and topology of $\Pi$ with respect to model parameters
    2. XI.2 Signatures to search for $F$ at the LHC
  12. XII Bounds on the model parameters
  13. XIII Summary and remarks

##  I Introduction

  * •

Theory motivation: Importance of LQ in general

  * •

Experimental motivation

    * –

B-physics anomalies

    * –

recent CMS results

  * •

NJL-SheSheng model

  * •

Highlight of this work

    * –

New mechanisms

    * –

Lepton pdf signatures investigating the origin of the lepton and the second
lepton in the event

    * –

physics reach (to be done)

    * –

final states with tau (to be done)

  * •

Impact to LHC physics

##  II LQ from four-fermion operators and contact interactions

  * •

Composite bosons

  * •

How LQ arise in the model

  * •

Comment from SheSheng ”SheSheng made a nice observation about the coupling to
mixed fermion generation, in analogy to the Standard Model, that lead to
define more appropriately the coupling g’ instead of g. We think this part
deserves to be reported in the paper, even though it is very advanced for its
implementation in an experimental search.”

  * •

Comment from SheSheng ”parameter space of Yukawa coupling $g_{Y}$ and mass
$M\\_pi$ for investigating for Pi0,+/- is the same as that of composite boson
with fraction charge (leptoquark). Also the theoretical calculations are the
same, what is changing is only the PDF, and final states. For Pi0,+/-, there
are channels of Pi0,+/- decaying to (produced by ) SM gauge bosons W,Z and
photon. While LQ states can decay to or produced by gluons. ”

  * •

Types of LQ and couplings to lepton-quark pair

    * –

Comparison with other schemes used to classify LQ

##  III Signal implementation and generation

  * •

Signal implementation in madgraph

    * –

choice of couplings and relationship with lambda used in CMS/LHC

  * •

Signal generation in madgraph

  * •

$\sigma$ comparison with other CMS models

    * –

Highlight of the main mechanisms used at LHC

    * –

Do they have gauge interactions implemented?

    * –

comparison with the new mechanisms and why $\sigma$ is higher

    * –

kinematic limits for resonant production, not present in t-channel (importance
of t-channel) (Pair production exceeds the kinematic limit, probably because
of the bwcutoff. This should happen also for single LQ, above 12/13 TeV)

  * •

Comparison with lepton PDF results

    * –

Reproducing $\sigma$

    * –

Investigating lepton origin inside the proton

    * –

LUX, MRS pdf discussion and results (discussion on leptons from Z and
neutrinos PDF) (see Sehar, 5 October)

    * –

New mechanisms with lepton PDF

      * *

all new mechanisms in link

      * *

t-channel tau-top

    * –

Further considerations. Comment ”from the point of view of the diagram, you
can also have neutrinos in the proton from QED and EW processes, so one can in
principle also consider the case of nu q -¿ LQ”

##  IV Phenomenology of LQ at the LHC

  * •

Production mechanisms and decays

  * •

Cross-section and decay widths

  * •

Comparison with current LHC processes

  * •

For the photon-induced processes,

    * –

we consider only inelastic case

    * –

we will compare the sensitivity to LQ with and without requiring the extra jet
that emitted the photon, expecting the latter case to be higher Of course, we
will add a section to compare the xSec for elastic and inelastic and comparing
to lepton-PDF

  * •

Search strategy at LHC

##  V Sensitivity at LHC and HL-LHC

  * •

Background production

  * •

Signal selection

  * •

Sensitivity plots

##  VI Summary and remarks

##  VII Four-fermion operators and contact interactions

The SM parity-violating (chiral) gauge symmetries and spontaneous/explicit
breaking of these symmetries for the $W$ and $Z$ gauge boson masses and the
hierarchy pattern of fermion masses have been at the centre of conceptual
elaboration for possible scenarios beyond. The No-Go theorem [Nielsen:1981xu,
1981PhLB..105..219N] shows that as a renormalizable quantum field theory at
infrared (IR) low energies, the SM bilinear fermion Lagrangian at ultraviolet
(UV) cutoff $\Lambda_{\rm cut}$ suffers theoretical inconsistencies with SM
chiral gauge symmetries and fermion spectra at low energies. It implies that
the BSM Lagrangian at $\Lambda_{\rm cut}$ should have quadrilinear operators
[Xue:2016dpl, Xue2017]

$\displaystyle\sum_{f=1,2,3}G_{\rm
cut}\,\Big{[}\bar{\psi}^{f}_{{}_{L}}\psi^{f}_{{}_{R}}\bar{\psi}^{f}_{{}_{R}}\psi^{f}_{{}_{L}}\Big{]}_{q=0,-1,2/3,-1/3},\quad
G_{\rm cut}\propto\Lambda^{-2}_{\rm cut},$ (1)

which is SM gauge symmetric four-fermion interactions of left- and right-
handed fermions ($\psi^{f}_{L},\psi^{f}_{R}$) in the SM charge sector “$q$”
and flavour family “$f$”. They are four-fermion operators of Nambu-Jona-
Lasinio (NJL) or Einstein-Cartan type. Their origins at the scale
$\Lambda_{\rm cut}$ are due to unknown grand unified interaction or quantum
gravity dynamics.

The strong four-fermion coupling $g_{\rm cut}\propto G_{\rm
cut}\Lambda^{2}_{\rm cut}$ dynamics leads to composite bosons
$\Pi^{f}\sim(\bar{\psi}^{f}_{{}_{R}}\psi^{f}_{{}_{L}})$ and fermions
$F^{f}_{R}\sim\psi^{f}_{{}_{R}}(\bar{\psi}^{f}_{{}_{R}}\psi^{f}_{{}_{L}})\sim\psi^{f}_{{}_{R}}\Pi^{f}$
[Xue1996, Xue2000, Xue2000a]. They are represented by the mass (pole) and form
factor (residue) of propagators, behaving as elementary particles after form-
factor renormalisation. These composite particles carry SM quantum numbers and
couple to massless SM gauge bosons. Their massive spectra preserve SM gauge
symmetries, and constituents can be leptons, quarks or leptoquarks. There
exists a UV fixed point $g^{*}_{\rm cut}$ and scaling domain, which realises
an SM gauge symmetric, renormalizable bilinear and dimensional-4 Lagrangian of
composite bosons and fermions at the composite scale $\Lambda\ll\Lambda_{\rm
cut}$ [Xue2014, Xue2017]. As the energy scale decreases below $\Lambda$, the
effective theory undergoes [Xue2023] a phase transition of composite particles
decay into constituents and spontaneous SM symmetries breaking via the top-
quark channel $(\bar{t}_{L}t_{R}\bar{t}_{R}t_{L})$, realising the
$\bar{t}t$-composite Higgs boson at the electroweak scale $v\approx 246$ GeV
[Bardeen1990]. The scale $\Lambda\approx 5.1$ TeV is estimated by using the
top quark and Higgs boson masses and extrapolating the renormalization-group
solution to the high-energy regime [Xue2013].

In this section we describe the four-fermion operators and contact
interactions that are relevant for the study of the phenomenology of the
composite bosons at $pp$ or $ep$ collisions, including the LHC, which will be
detailed in Sec. X.

We consider, among four-fermion operators (1) in introduction, the following
SM gauge-symmetric and fermion-number conserving four-fermion operators,

$\displaystyle
G\left[(\bar{\ell}^{i}_{L}e_{R})(\bar{d}^{a}_{R}\psi_{Lia})+(\bar{\ell}^{i}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}\psi_{Lia})\right]+{\rm
h.c.},$ (2)

and

$\displaystyle
G\left[(\bar{\ell}^{i}_{L}\nu^{e}_{R})(\bar{\nu}^{e}_{R}\ell_{Li})+(\bar{\ell}^{i}_{L}e_{R})(\bar{e}_{R}\ell_{Li})\right]+{\rm
h.c.}$ $\displaystyle
G\left[(\bar{\ell}^{i}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\ell_{Li})+(\bar{\ell}^{i}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\ell_{Li})\right]+{\rm
h.c.}$ (3)

and

$\displaystyle
G\left[(\bar{\psi}_{Lia}\nu^{e}_{R})(\bar{\nu}^{e}_{R}\psi_{Lia})+(\bar{\psi}_{Lia}e_{R})(\bar{e}_{R}\psi_{Lia})\right]+{\rm
h.c.},$ (4) $\displaystyle
G\left[(\bar{\psi}^{bi}_{L}u_{Rb})(\bar{u}^{a}_{R}\psi_{Lia})+(\bar{\psi}^{bi}_{L}d_{Rb})(\bar{d}^{a}_{R}\psi_{Lia})\right]+{\rm
h.c.},$ (5)

being the SM doublet $\ell^{i}_{L}=(\nu^{e}_{L},e_{L})$ and singlet $e_{R}$
with an additional right-handed neutrino $\nu^{e}_{R}$ for leptons;
$\psi_{Lia}=(u_{La},d_{La})$ and $u^{a}_{R},d^{a}_{R}$ for quarks, where the
color $a,b$ and $SU_{L}(2)$-isospin $i$ indexes are summed over. Eqs.
(146)-(145) are for the first family only, as a representative of the three
fermion families. The SM left- and right-handed fermions are mass eigenstates,
their masses are negligible in TeV-energy regime and small mixing among three
families encoded in $G$ is also neglected [31].

From Eqs. (146), (3) and (4), we give explicitly quark-lepton interactions of
three types,

$\displaystyle
G\left[(\bar{e}_{L}e_{R})(\bar{d}^{a}_{R}d_{La})+(\bar{\nu}^{e}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}u_{La})+(\bar{\nu}^{e}_{L}e_{R})(\bar{d}^{a}_{R}u_{La})\right]+{\rm
h.c.},$ (6) $\displaystyle
G\left[(\bar{\nu}^{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\nu^{e}_{L})+(\bar{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}e_{L})+(\bar{\nu}^{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\nu^{e}_{L})+(\bar{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}e_{L})\right]+{\rm
h.c.},$ $\displaystyle
G\left[(\bar{u}^{a}_{L}\nu^{e}_{R})(\bar{\nu}^{e}_{R}u^{a}_{L})+(\bar{d}^{a}_{L}\nu^{e}_{R})(\bar{\nu}^{e}_{R}d^{a}_{L})+(\bar{u}^{a}_{L}e_{R})(\bar{e}_{R}u^{a}_{L})+(\bar{d}^{a}_{L}e_{R})(\bar{e}_{R}d^{a}_{L})\right]+{\rm
h.c.},$

for the first generation. $\nu^{e},e,u,d\rightarrow\nu^{\mu},\mu,c,s$ for the
second generation, and $\nu^{e},e,u,d\rightarrow\nu^{\tau},\tau,t,b$ for the
third generation.

Figure 1: We show the Feynman diagrammatic representation for the contact
interaction between the composite boson and its constituent quarks, where the
thin solid line represents an SM elementary fermion, double wave line
represents a composite boson, and the blob represents an interacting vertex
$(F_{\Pi}/\Lambda)^{2}P_{L,R}$, where $P_{L,R}=(1\mp\gamma_{5})/2$.

###  VII.1 Composite bosons $\Pi^{0,\pm}$

Composite bosons $\Pi$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$\Pi^{+}$ | $(\bar{d}^{a}_{R}u_{La})$ | $+1$ | $~{}~{}1/2$ | $1/2$
$\Pi^{-}$ | $(\bar{u}^{a}_{R}d_{La})$ | $-1$ | $-1/2$ | $-1/2$
$\Pi^{0}_{u}$ | $(\bar{u}^{a}_{R}u_{La})$ | $0$ | $~{}~{}1/2$ | $-1/2$
$\Pi^{0}_{d}$ | $(\bar{d}^{a}_{R}d_{La})$ | $0$ | $-1/2$ | $1/2$
Table 1: Composite bosons $\Pi^{0,\pm}$, their constituents and SM quantum
numbers. The color $a$ index is summed.

From the four-fermion interaction in Eq. (146) or (145), it is possible to
form charged composite bosons

$\displaystyle\Pi^{+}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}(\bar{d}^{a}_{R}u_{La}),$ (7)
$\displaystyle\Pi^{-}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}(\bar{u}^{a}_{R}d_{La}),$ (8)

$\Pi^{-}=(\Pi^{+})^{\dagger}$, and neutral composite bosons

$\displaystyle\Pi^{0}_{u}=(g^{*}/F_{\Pi})^{2}(\bar{u}^{a}_{R}u_{La}),$ (9)
$\displaystyle\Pi^{0}_{d}=(g^{*}/F_{\Pi})^{2}(\bar{d}^{a}_{R}d_{La}),$ (10)

and their Hermitian conjugates. Such normalized composite boson field has the
same dimension $[energy]$ of elementary boson field. The composite boson
carries the quantum numbers that are the sum over SM quantum numbers of its
two constituents, see Table 1. Composite bosons $(\Pi^{+},\Pi^{-})$ and
$(\Pi^{0}_{u},\Pi^{0}_{d})$ form $SU_{L}(2)$ isoposin doublets, and
$SU_{c}(3)$ colour singlets. Their wavefunctions are $S$-wave of antiparallel
spin arrangement $(\uparrow\downarrow+\downarrow\uparrow)$ and zero orbital
angular momentum. These are pseudo composite bosons $\Pi^{0,\pm}$, analogous
to charged and neutral pions $\pi^{0,\pm}$ in the low-energy QCD.

###  VII.2 Quark-lepton composite bosons $\Pi^{5/3}_{a}$, $\Pi^{-1/3}_{a}$ and
$\Pi^{2/3}_{u_{a}}$, $\Pi^{2/3}_{d_{a}}$

LQ Composite bosons $\Pi_{a}^{Q}$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$\Pi^{+5/3}_{a}$ | $(\bar{e}_{R}u_{La})$ | $+5/3$ | $~{}~{}1/2$ | $7/6$
$\Pi^{-1/3}_{a}$ | $(\bar{\nu}^{e}_{R}d_{La})$ | $-1/3$ | $-1/2$ | $1/6$
$\Pi^{2/3}_{u_{a}}$ | $(\bar{\nu}^{e}_{R}u_{La})$ | $2/3$ | $~{}~{}1/2$ | $1/6$
$\Pi^{2/3}_{d_{a}}$ | $(\bar{e}_{R}d_{La})$ | $2/3$ | $-1/2$ | $7/6$
Table 2: Quark-lepton composite bosons $\Pi^{5/3}_{a}$, $\Pi^{-1/3}_{a}$, $\Pi^{2/3}_{u_{a}}$ and $\Pi^{2/3}_{d_{a}}$ their constituents and SM quantum numbers. They are colour states and the colour $a$ index is not summed. LQ Composite bosons $\Pi_{a}^{Q}$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$\Pi^{-5/3}_{a}$ | $(\bar{u}_{Ra}e_{L})$ | $-5/3$ | $~{}~{}-1/2$ | $-7/6$
$\Pi^{+1/3}_{a}$ | $(\bar{d}_{Ra}\nu^{e}_{L})$ | $1/3$ | $1/2$ | $-1/6$
$\Pi^{-2/3}_{u_{a}}$ | $(\bar{u}_{Ra}\nu^{e}_{L})$ | $-2/3$ | $~{}~{}1/2$ | $-7/6$
$\Pi^{-2/3}_{d_{a}}$ | $(\bar{d}_{Ra}e_{L})$ | $-2/3$ | $-1/2$ | $-1/6$
Table 3: Quark-lepton composite bosons $\Pi^{-5/3}_{a}$, $\Pi^{1/3}_{a}$,
$\Pi^{-2/3}_{u_{a}}$ and $\Pi^{-2/3}_{d_{a}}$ their constituents and SM
quantum numbers. They are colour states and the colour $a$ index is not
summed. $(\Pi^{5/3}_{a})^{\dagger}=\Pi^{-5/3}_{a}$,
$(\Pi^{-1/3}_{a})^{\dagger}=\Pi^{1/3}_{a}$ and
$(\Pi^{2/3}_{u_{a}})^{\dagger}=\Pi^{-2/3}_{d_{a}}$,
$(\Pi^{2/3}_{d_{a}})^{\dagger}=\Pi^{-2/3}_{u_{a}}$. They are conjugated fields
of those in Table 3. We will indicate these four LQ composite bosons by
$\Pi_{a}^{Q}$.

From the four-fermion interaction in Eq. (146), it is also possible to form
the following charged quark-lepton composite bosons of four kinds

$\displaystyle\Pi^{+5/3}_{a}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}(\bar{e}_{R}u_{La}),\quad\Pi^{-5/3}_{a}=(g^{*}/F_{\Pi})^{2}(\bar{u}_{Ra}e_{L}),$
(11) $\displaystyle\Pi^{-1/3}_{a}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}(\bar{\nu}^{e}_{R}d_{La}),\quad\Pi^{+1/3}_{a}=(g^{*}/F_{\Pi})^{2}(\bar{d}_{Ra}\nu^{e}_{L}),$
(12)

and

$\displaystyle\Pi^{2/3}_{u_{a}}=(g^{*}/F_{\Pi})^{2}(\bar{\nu}^{e}_{R}u_{La}),\quad\Pi^{-2/3}_{u_{a}}=(g^{*}/F_{\Pi})^{2}(\bar{u}_{Ra}\nu^{e}_{L}),$
(13)
$\displaystyle\Pi^{2/3}_{d_{a}}=(g^{*}/F_{\Pi})^{2}(\bar{e}_{R}d_{La}),\quad\Pi^{-2/3}_{d_{a}}=(g^{*}/F_{\Pi})^{2}(\bar{d}_{Ra}e_{L}),$
(14)

and their Hermitian conjugates in the right. Such normalized composite boson
field has the same dimension $[energy]$ of elementary boson field. The quark-
lepton composite bosons carry the quantum numbers that are the sum over SM
quantum numbers of its two constituents, see Table 2. Quark-lepton composite
bosons ($\Pi^{5/3}_{a}$, $\Pi^{-1/3}_{a}$) and
($\Pi^{2/3}_{u_{a}},\Pi^{2/3}_{d_{a}})$ form $SU_{L}(2)$ isopospin doublets,
and $SU_{c}(3)$ colour states. Their wave functions are $S$-wave of
antiparallel spin arrangement $\uparrow\downarrow$ or $\downarrow\uparrow$ and
zero orbital angular momentum. These pseudo quark-lepton composite bosons have
no counterparts in meson spectra in the low-energy QCD.

The quark-lepton composite bosons (11-14) and composite bosons (7-10) are
related by

$\displaystyle e_{R}\iff d_{R},\quad\nu^{e}_{R}\iff u_{R}.$ (15)

Quarl-lepton composite bosons can be diagrammatically represented by Fig. 3 by
replacing one of quarks $q$ to a lepton $\ell$ and replacing $\Pi$ to
$\Pi_{a}$.

###  VII.3 Contact interaction of composite bosons, quarks and leptons

As shown in Fig. 3, the effective coupling between composite boson and its two
constituents can be written as an effective contact interaction,

$\displaystyle{\mathcal{L}}^{\rm\Pi^{\pm}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm Y}}(\bar{d}^{a}_{R}u_{La})\Pi^{-}+{\rm h.c.},$ (16)
$\displaystyle{\mathcal{L}}^{\rm\Pi^{0}_{d}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm Y}}(\bar{d}^{a}_{R}d_{La})\Pi^{0}_{d}+{\rm h.c.},$
(17) $\displaystyle{\mathcal{L}}^{\rm\Pi^{0}_{u}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm Y}}(\bar{u}^{a}_{R}u_{La})\Pi^{0}_{u}+{\rm h.c.},$
(18)

and

$\displaystyle{\mathcal{L}}^{\rm\Pi^{+5/3}_{a}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm Y}}(\bar{e}_{R}u_{La})\Pi^{+5/3}_{a}+{\rm h.c.},$
(19) $\displaystyle{\mathcal{L}}^{\rm\Pi^{-1/3}_{a}}_{\rm CI}$
$\displaystyle=$ $\displaystyle g_{{}_{\rm
Y}}(\bar{\nu}^{e}_{R}d_{La})\Pi^{-1/3}_{a}+{\rm h.c.},$ (20)
$\displaystyle{\mathcal{L}}^{\rm\Pi^{2/3}_{u_{a}}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm Y}}(\bar{\nu}^{e}_{R}u_{La})\Pi^{2/3}_{u_{a}}+{\rm
h.c.},$ (21) $\displaystyle{\mathcal{L}}^{\rm\Pi^{2/3}_{d_{a}}}_{\rm CI}$
$\displaystyle=$ $\displaystyle g_{{}_{\rm
Y}}(\bar{e}_{R}d_{La})\Pi^{2/3}_{d_{a}}+{\rm h.c.},$ (22)

where $g_{{}_{\rm Y}}=(F_{\Pi}/\Lambda)^{2}$. Appropriate normalizing the
composite boson $\Pi$ with the form factor $(g^{*}/F_{\Pi})^{2}$ in Eqs.
(7-10), the effective contact interaction in Eqs. (16-18) and (19-22) can be
expressed as a dimensionless Yukawa coupling $g_{{}_{\rm Y}}$, whose value,
corresponding to $F_{\Pi}$ value, can be different for composite bosons in
Eqs. (7-10) and (11-13), but we do not consider such difference here. These
are relevant contact interactions for phenomenological studies of possible
inelastic channels of composite-fermion production and decay in $pp$ or $ep$
collisions.

We discuss the issue of generation mixing for four different charged types of
LQ states: $\Pi^{5/3}_{a}$, $\Pi^{-1/3}_{a}$ $\Pi^{2/3}_{u_{a}}$ and
$\Pi^{2/3}_{d_{a}}$ in Table 2. Each type has three copies due to generations,
taking as example $\Pi^{5/3}_{a}\sim(\bar{e}_{R}u_{La})$,
$\Pi^{5/3}_{a}\sim(\bar{\mu}_{R}c_{La})$ and
$\Pi^{5/3}_{a}\sim(\bar{\tau}_{R}t_{La})$. If the LQ $\Pi^{5/3}_{a}$ is
produced by, then decay to its constituents in the same generation, the Yukawa
coupling $g_{Y}\sim{\mathcal{O}}(1)$ as given in Eqs.(19-22). Now we consider
$\Pi^{5/3}_{a}$ is produced by $e$ and $u$ of the first family, and the
branching ratio of decaying to $\tau$ and $t$ of the third family. The Yukawa
coupling $g_{Y}\sim{\mathcal{O}}(1)$ for $e$ and $u$ production, and another
Yukawa coupling $g^{\prime}_{Y}$ for decaying to $\tau$ and $t$. The Yukawa
coupling $g^{\prime}_{Y}$ is a new parameter and $g^{\prime}_{Y}<g_{Y}$,
possibly $g^{\prime}_{Y}\ll g_{Y}$ for the following reasons. The left- and
right-handed fields of leptons and quarks, which make composite bosons and
fermions, are the SM gauge eigenstates in each generation. They are the same
as the mass eigenstates without considering generation mixing. If we need to
consider the mixing, see the unitary flavour transformations eqs (15,16) for
quarks and eqs (23, 24) for leptons in my article JHEP 2016 ref.
https://arxiv.org/pdf/1605.01266.pdf. Using these unitary transformations
$U^{\nu,\ell,u,d}_{L,R}$, we can rewrite the Yukawa couplings (16)-(22) by
associating two unitary matrices $(U)^{\dagger}U$ which are unknown. As
example, by interacting (19) the $e$ and $u$ produced $\Pi_{a}^{5/3}$ will
decay to $\tau$ and $t$ by interacting

$\displaystyle{\mathcal{L}}^{\rm\Pi^{+5/3}_{a}}_{\rm CI}$ $\displaystyle=$
$\displaystyle g_{{}_{\rm
Y}}(U^{\dagger}_{R}U_{L})(\bar{\tau}_{R}t_{La})\Pi^{+5/3}_{a}+{\rm
h.c.}=g^{\prime}_{{}_{\rm Y}}(\bar{\tau}_{R}t_{La})\Pi^{+5/3}_{a}+{\rm h.c.},$
(23)

where $g^{\prime}_{{}_{\rm Y}}=g_{{}_{\rm Y}}(U^{\dagger}_{R}U_{L})$, and the
notation $(U^{\dagger}_{R}U_{L})$ stands for relevant unitary transformations
between generations considered. The general mixing matrix
$(U^{\dagger}_{R}U_{L})$ of leptoquark sector, see eq (35) of my JHEP 2016,
has its analogies CKM matrix in quark sector, PMNS matrix in lepton sector. In
conclusion, the general mixing matrix $(U^{\dagger}_{R}U_{L})$ is totally
unknown, the effective way for dealing with composite states of different
generations but the same SM quantum numbers is to introduce a new Yukawa
coupling. These discussions also apply to the three nonLQ composite bosons
$\Pi^{-}$, $\Pi^{0}_{u}$ and $\Pi^{0}_{d}$ see eqs. (16)-(17). It can be
considered as a general rule for considering the initial and final states are
not in the same family.

##  VIII LQ Composite fermions $F_{a}$

There are three types operators (146), (3) and (4) that can form LQ composite
fermion states.

###  VIII.1 The type 1 of
$G\left[(\bar{\ell}^{i}_{L}e_{R})(\bar{d}^{a}_{R}\psi_{Lia})+(\bar{\ell}^{i}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}\psi_{Lia})\right]+{\rm
h.c.}$ (146)

In Eqs. (146), each four-fermion operator has the two possibilities to form
composite fermions, listed in Tables 4 and 5. Up to a form factor, $E$ ($N$)
indicate a composite fermion composed by an electron (a neutrino) and a
colorless quark pair $\Pi^{0,\pm}$, and its superscript for electric charge.
In Eq. (146), there are four independent composite fields $F$: $N_{R}^{0}$,
$N_{R}^{-}$, $E^{0}_{R}$, $E^{-}_{R}$, and their Hermitian conjugates:
$\bar{N}^{0}_{L}=(N_{R}^{0})^{\dagger}\gamma_{0}$,
$\bar{N}^{+}_{La}=(N_{Ra}^{-})^{\dagger}\gamma_{0}$,
$\bar{E}^{0}_{La}=(E^{0}_{Ra})^{\dagger}\gamma_{0}$,
$\bar{E}^{+}_{L}=(E^{-}_{R})^{\dagger}\gamma_{0}$. See tables 4 and 5. They
carry SM quantum numbers $t^{i}_{3L},Y$, and $Q_{i}=Y+t^{i}_{3L}$, which are
the sum of SM quantum numbers ($t^{i}_{3L},Y,Q_{i}$) of their constituents,
i.e., the elementary leptons and quarks in the same SM family [32], listed in
Tables 4 and 5, so that the contact interactions in Eq. (99) are SM gauge
symmetric.

The four composite fields $F$ (colorless): $N_{R}^{0}$, $N_{R}^{-}$,
$E^{0}_{R}$, $E^{-}_{R}$, are two-component right-handed Wely fermions.

They form two right-handed SU(2) doublets (colorless): (i)
$N_{R}=(N_{R}^{0},N_{R}^{-})=\Pi_{-1/2}\nu^{e}_{R}$ of u channel hypercharge
-1/2 +0= -1/2, (ii) $E_{R}=(E_{R}^{0},E_{R}^{-})=\Pi_{1/2}e_{R}$ of d channel
hypercharge 1/2 -1= -1/2, where the SU(2) composite boson doublets:
$\Pi_{-1/2}=(\Pi^{0}_{u},\Pi^{-})$ of u channel the hypercharge -1/2, and
$\Pi_{1/2}=(\Pi^{+},\Pi^{0}_{d})$ of d channel,the hypercharge 1/2.

The SU(2) doublet massive Dirac fermions (four spinor components) are: (i)
$N_{D}=(N_{R}^{0}+\nu^{e}_{L},N_{R}^{-}+e_{L})$ of hypercharge -1/2 and Dirac
mass term $M_{F}\bar{N}_{D}N_{D}$; (ii)
$E_{D}=(E_{R}^{0}+\nu^{e}_{L},E_{R}^{-}+e_{L})$ of hypercharge -1/2 and Dirac
mass term $M_{F}\bar{E}_{D}E_{D}$.

There are two left-handed SU(2) singlets: (i)
$N^{s}_{L}=\Pi^{\dagger}_{-1/2}(\nu^{e}_{L},e_{L})$ of u channel of
hypercharge 1/2-1/2=0, (ii) $E^{s}_{L}=\Pi^{\dagger}_{1/2}(\nu^{e}_{L},e_{L})$
of d channel of hypercharge -1/2-1/2=-1, where the SU(2) composite boson
doublets: $\Pi^{\dagger}_{-1/2}=(\Pi^{0}_{u},\Pi^{-})^{\dagger}$ of u channel
the hypercharge 1/2, and $\Pi^{\dagger}_{1/2}=(\Pi^{+},\Pi^{0}_{d})^{\dagger}$
of d channel,the hypercharge -1/2.

The SU(2) singlet massive Dirac fermions (four spinor components) are: (i)
$N^{s}_{D}=(N^{s}_{L}+\nu^{e}_{R})$ of hypercharge 0, electrically neutral,
and Dirac mass term $M_{F}\bar{N}_{D}N_{D}$; (ii)
$E^{s}_{D}=(E^{s}_{L}+e_{R})$ of hypercharge -1, charge -1 and Dirac mass term
$M_{F}\bar{E}_{D}E_{D}$.

Operator | LQ Composite fermion $F_{R}$ | LQ Composite fermion $\bar{F}_{L}$ | Composite boson $\Pi$
---|---|---|---
$(\bar{\nu}^{e}_{L}\nu^{e}_{R})(\bar{u}_{Ra}u_{La})$ | $N^{0}_{R}\sim\nu^{e}_{R}(\bar{u}_{Ra}u_{La})$ | $\bar{N}^{0}_{L}\sim\bar{\nu}^{e}_{L}(\bar{u}^{a}_{R}u_{La})$ | $\Pi^{0}_{u}\sim(\bar{u}_{Ra}u_{La})$
$(\bar{e}_{L}\nu^{e}_{R})(\bar{u}_{Ra}d_{La})$ | $N^{-}_{R}\sim\nu^{e}_{R}(\bar{u}_{Ra}d_{La})$ | $\bar{N}^{+}_{L}\sim\bar{\nu}^{e}_{L}(\bar{d}^{a}_{R}u_{La})$ | $\Pi^{-}_{a}\sim(\bar{u}_{Ra}d_{La})$
$(\bar{\nu}^{e}_{L}e_{R})(\bar{d}_{Ra}u_{La})$ | $E^{0}_{R}\sim e_{R}(\bar{d}_{Ra}u_{La})$ | $\quad\bar{E}^{0}_{L}\sim\bar{e}_{L}(\bar{u}^{a}_{R}d_{aL})$ | $\Pi^{+}\sim(\bar{d}_{Ra}u_{La})$
$(\bar{e}_{L}e_{R})(\bar{d}_{Ra}d_{La})$ | $E^{-}_{R}\sim e_{R}(\bar{d}_{Ra}d_{La})$ | $\bar{E}^{+}_{L}\sim\bar{e}_{L}(\bar{d}^{a}_{R}d_{La})$ | $\Pi^{0}_{d}\sim(\bar{d}_{Ra}d_{La})$
Table 4: Four-fermion operators (146) and possible colorless LQ composite fermions $F$. The color $a$ index is summed, these composite fermions are colorless, final states are two quarks and one lepton. We consider them as LQ composite fermions. These composite fermions are the same as $N(neutrino)$ and $E(electron)$ sector table I and III in previous Roberto article. LQ composite fermions $F_{R}$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$N^{0}_{R}$ | $\nu^{e}_{R}(\bar{u}_{Ra}u_{La})$ | $0$ | $~{}~{}1/2$ | $-1/2$
$N^{-}_{R}$ | $\nu^{e}_{R}(\bar{u}_{Ra}d_{La})$ | $-1$ | $-1/2$ | $-1/2$
$E^{0}_{R}$ | $e_{R}(\bar{d}_{Ra}u_{La})$ | $0$ | $~{}~{}1/2$ | $-1/2$
$E^{-}_{R}$ | $e_{R}(\bar{d}_{Ra}d_{La})$ | $-1$ | $-1/2$ | $-1/2$
Table 5: LQ Composite fermions $F_{R}$ are colorless, their constituents and
SM quantum numbers, corresponding to Table 4. These composite fermions are the
same as $N$ and $E$ sector table I and III in previous Roberto article.

###  VIII.2 The type 3 of
$G\left[(\bar{\psi}_{Lia}\nu^{e}_{R})(\bar{\nu}^{e}_{R}\psi_{Lia})+(\bar{\psi}_{Lia}e_{R})(\bar{e}_{R}\psi_{Lia})\right]+{\rm
h.c.}$ (4)

Operator | LQ Composite fermion $F_{R}$ | LQ Composite fermion $\bar{F}_{L}$ | Composite boson $\Pi$
---|---|---|---
$(\bar{u}_{La}\nu^{e}_{R})(\bar{\nu}^{e}_{R}u_{La})$ | $N^{2/3}_{Ra}\sim\nu^{e}_{R}(\bar{\nu}^{e}_{R}u_{La})$ | $\bar{N}^{-2/3}_{La}\sim\bar{\nu}^{e}_{L}(\bar{u}^{a}_{R}\nu^{e}_{L})$ | $\Pi^{2/3}_{u_{a}}\sim(\bar{\nu}^{e}_{R}u_{La})$
$(\bar{d}_{La}\nu^{e}_{R})(\bar{\nu}^{e}_{R}d_{La})$ | $N^{-1/3}_{Ra}\sim\nu^{e}_{R}(\bar{\nu}^{e}_{R}d_{La})$ | $\bar{N}^{1/3}_{La}\sim\bar{\nu}^{e}_{L}(\bar{d}^{a}_{R}\nu^{e}_{L})$ | $\Pi^{-1/3}_{a}\sim(\bar{\nu}^{e}_{R}d_{La})$
$(\bar{u}_{La}e_{R})(\bar{e}_{R}u_{La})$ | $E^{2/3}_{Ra}\sim e_{R}(\bar{e}_{R}u_{La})$ | $\quad\bar{E}^{-2/3}_{La}\sim\bar{e}_{L}(\bar{u}^{a}_{R}e_{L})$ | $\Pi^{5/3}_{a}\sim(\bar{e}_{R}u_{La})$
$(\bar{d}_{La}e_{R})(\bar{e}_{R}d_{La})$ | $E^{-1/3}_{Ra}\sim e_{R}(\bar{e}_{R}d_{La})$ | $\bar{E}^{1/3}_{La}\sim\bar{e}_{L}(\bar{d}^{a}_{R}e_{L})$ | $\Pi^{2/3}_{d_{a}}\sim(\bar{e}_{R}d_{La})$
Table 6: Four-fermion operators (4) and possible LQ composite fermions $F$.
The color $a$ index is not summed, these composite fermions carry color,
couple to gluons and can be created by two gluons collisions, final states are
two leptons and one quark. This property is the same as $U$ and $D$ sector in
previous article, however the final state is three quarks.

In Eqs. (4), each four-fermion operator has the two possibilities to form
composite fermions, listed in Tables 6 and 7. Up to a form factor, $E$ ($N$)
indicate a composite fermion composed by an electron (a neutrino) and a color
lepton quark pair $\Pi_{a}$, and its superscript for electric charge. In Eq.
(4), there are four independent composite fields $F$: $N_{Ra}^{2/3}$,
$N_{Ra}^{-1/3}$, $E^{2/3}_{Ra}$, $E^{-1/3}_{Ra}$ and their Hermitian
conjugates: $\bar{N}^{-2/3}_{La}=(N_{Ra}^{2/3})^{\dagger}\gamma_{0}$,
$\bar{N}^{1/3}_{La}=(N_{Ra}^{-1/3})^{\dagger}\gamma_{0}$,
$\bar{E}^{-2/3}_{La}=(E^{2/3}_{Ra})^{\dagger}\gamma_{0}$,
$\bar{E}^{1/3}_{La}=(E^{-1/3}_{Ra})^{\dagger}\gamma_{0}$. See tables 6 and 7.
They carry SM quantum numbers $t^{i}_{3L},Y$, and $Q_{i}=Y+t^{i}_{3L}$, which
are the sum of SM quantum numbers ($t^{i}_{3L},Y,Q_{i}$) of their
constituents, i.e., the elementary leptons and quarks in the same SM family
[32], listed in Table 9 and 7, so that the contact interactions in Eq.
(LABEL:c1) are SM gauge symmetric.

The four composite fields $F$: $N_{Ra}^{2/3}$, $N_{Ra}^{-1/3}$,
$E^{2/3}_{Ra}$, $E^{-1/3}_{Ra}$ are two-component right-handed Wely fermions.

They form two right-handed SU(2) doublets: (i)
$N_{Ra}=(N^{2/3}_{Ra},N^{-1/3}_{Ra})=\Pi^{a}_{1/6}\nu^{e}_{R}$ of u channel
hypercharge 1/6 +0= 1/6, (ii)
$E_{Ra}=(E^{2/3}_{Ra},E^{-1/3}_{Ra})=\Pi_{7/6}e_{R}$ of d channel hypercharge
7/6 -1= 1/6, where the SU(2) LQ composite boson doublets:
$\Pi^{a}_{1/6}=(\Pi^{2/3}_{u_{a}},\Pi^{-1/3}_{a})$ of u channel the
hypercharge 1/6, and $\Pi^{a}_{7/6}=(\Pi^{5/3}_{a},\Pi^{2/3}_{d_{a}})$ of d
channel the hypercharge 7/6.

The SU(2) doublet massive Dirac fermions (four spinor components) are: (i)
$N^{a}_{D}=(N^{2/3}_{Ra}+u_{La},N^{-1/3}_{Ra}+d_{La})$ of hypercharge 1/6 and
Dirac mass term $M_{F}\bar{N}_{D}N_{D}$; (ii)
$E^{a}_{D}=(E^{2/3}_{Ra}+u_{La},E^{-1/3}_{Ra}+d_{La})$ of hypercharge 1/6 and
Dirac mass term $M_{F}\bar{E}_{D}E_{D}$.

There are two left-handed SU(2) singlets: (i)
$N^{s}_{La}=(\nu^{e}_{L},e_{L})\Pi^{a}_{1/6}$ of u channel of hypercharge
1/6-1/2=-1/3= charge, (ii) $E^{s}_{La}=(\nu^{e}_{L},e_{L})\Pi^{a}_{7/6}$ of d
channel of hypercharge 7/6-1/2=2/3= charge, where the SU(2) LQ composite boson
doublets: $\Pi^{a}_{1/6}=(\Pi^{2/3}_{u_{a}},\Pi^{-1/3}_{a})$ of u channel the
hypercharge 1/6, and $\Pi^{a}_{7/6}=(\Pi^{5/3}_{a},\Pi^{2/3}_{d_{a}})$ of d
channel the hypercharge 7/6.

The SU(2) singlet massive Dirac fermions (four spinor components) are: (i)
$N^{s}_{Da}=(N^{s}_{La}+d_{Ra})$ of hypercharge -1/3, charge e=Y=-1/3 and
Dirac mass term $M_{F}\bar{N}_{Da}N_{Da}$; (ii)
$E^{s}_{Da}=(E^{s}_{La}+u_{Ra})$ of hypercharge 2/3, charge e=Y=2/3 and Dirac
mass term $M_{F}\bar{E}_{Da}E_{Da}$.

LQ composite fermions $F_{R}$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$N^{2/3}_{Ra}$ | $\nu^{e}_{R}(\bar{\nu}^{e}_{R}u_{La})$ | $2/3$ | $~{}~{}1/2$ | $1/6$
$N^{-1/3}_{Ra}$ | $\nu^{e}_{R}(\bar{\nu}^{e}_{R}d_{La})$ | $-1/3$ | $-1/2$ | $1/6$
$E^{2/3}_{Ra}$ | $e_{R}(\bar{e}_{R}u_{La})$ | $2/3$ | $~{}~{}1/2$ | $1/6$
$E^{-1/3}_{Ra}$ | $e_{R}(\bar{e}_{R}d_{La})$ | $-1/3$ | $-1/2$ | $1/6$
Table 7: LQ Composite fermions $F_{R}$, coupling to gluons, their constituents and SM quantum numbers, corresponding to Table 6 Operator | Composite fermion $F_{R}$ | Composite fermion $\bar{F}_{L}$ | Composite boson $\Pi$
---|---|---|---
$(\bar{\nu}^{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\nu^{e}_{L})$ | $U^{0}_{R}\sim u^{a}_{R}(\bar{u}^{a}_{R}\nu^{e}_{L})$ | $\quad\bar{U}^{0}_{L}\sim\bar{u}^{a}_{L}(\bar{\nu}^{e}_{R}u_{La})$ | $\Pi^{-2/3}_{u_{a}}\sim(\bar{u}^{a}_{R}\nu^{e}_{L})$
$(\bar{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}e_{L})$ | $U^{-}_{R}\sim u^{a}_{R}(\bar{u}^{a}_{R}e_{L})$ | $\bar{U}^{+}_{L}\sim\bar{u}^{a}_{L}(\bar{e}_{R}u_{La})$ | $\Pi^{-5/3}_{a}\sim(\bar{u}^{a}_{R}e_{L})$
$(\bar{\nu}^{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\nu^{e}_{L})$ | $D^{0}_{R}\sim d^{a}_{R}(\bar{d}^{a}_{R}\nu^{e}_{L})$ | $\bar{D}^{0}_{L}\sim\bar{d}^{a}_{L}(\bar{\nu}^{e}_{L}d^{a}_{R})$ | $\Pi^{+1/3}_{a}\sim(\bar{d}^{a}_{R}\nu^{e}_{L})$
$(\bar{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}e_{L})$ | $D^{-}_{R}\sim d^{a}_{R}(\bar{d}^{a}_{R}e_{L})$ | $\bar{D}^{+}_{L}\sim\bar{d}^{a}_{L}(\bar{e}_{R}d_{La})$ | $\Pi^{-2/3}_{d_{a}}\sim(\bar{d}^{a}_{R}e_{L})$
Table 8: Four-fermion operators in Eq. (3) and possible composite fermions $F$ and composite bosons $\Pi_{a}$. The repeat color $a$ index is summed, these $F$ do not couple to gluons at leading order. LQ composite fermions $F_{R}$ | constituents | charge $Q_{i}=Y+t^{i}_{3L}$ |  $SU_{L}(2)$ 3-isospin $t^{i}_{3L}$ |  $U_{Y}(1)$-hypercharge $Y$
---|---|---|---|---
$U^{0}_{R}$ | $u^{a}_{R}(\bar{u}^{a}_{R}\nu^{e}_{L})$ | $0$ | $~{}~{}1/2$ | $-1/2$
$U^{-}_{R}$ | $u^{a}_{R}(\bar{u}^{a}_{R}e_{L})$ | $-1$ | $-1/2$ | $-1/2$
$D^{0}_{R}$ | $d^{a}_{R}(\bar{d}^{a}_{R}\nu^{e}_{L})$ | $0$ | $~{}~{}1/2$ | $-1/2$
$D^{-}_{R}$ | $d^{a}_{R}(\bar{d}^{a}_{R}e_{L})$ | $-1$ | $-1/2$ | $-1/2$
Table 9: LQ Composite fermions $F_{R}$, their constituents and SM quantum
numbers. They come from strong four-fermion interaction (3). These have the
same SM quantum numbers as the $E,N$ sector in previous article, however,
different final states. There are two constituent quarks, and the repeat color
index is summed, these composite fermions have no color, and do not couple to
gluons at leading order. The $E,N$ sector in previous article also does not
couple to gluons

###  VIII.3 The type 2 of
$G\left[(\bar{\ell}^{i}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\ell_{Li})+(\bar{\ell}^{i}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\ell_{Li})\right]+{\rm
h.c.}$ (3)

In Eqs. (3), each four-fermion operator has the two possibilities to form
composite fermions, listed in Tables 8 and 9. Up to a form factor, $D$ ($U$)
indicate a composite fermion composed by a down quark $d$ (an up quark $u$)
and a color lepton quark pair $\Pi_{a}$, and its superscript for electric
charge. In Eq. (3), there are four independent composite fields $F$:
$U^{0}_{R}$, $U_{R}^{-}$, $D_{R}^{0}$, $D^{-}_{R}$ and their Hermitian
conjugates: $\bar{U}^{0}_{L}=(U^{0}_{R})^{\dagger}\gamma_{0}$,
$\bar{U}^{+}_{L}=(U^{-}_{R})^{\dagger}\gamma_{0}$,
$\bar{D}^{0}_{L}=(D^{0}_{R})^{\dagger}\gamma_{0}$,$\bar{D}^{+}_{L}=(D^{-}_{R})^{\dagger}\gamma_{0}$,
. See tables 8 and 9. They carry SM quantum numbers $t^{i}_{3L},Y$, and
$Q_{i}=Y+t^{i}_{3L}$, which are the sum of SM quantum numbers
($t^{i}_{3L},Y,Q_{i}$) of their constituents, i.e., the elementary leptons and
quarks in the same SM family [32], so that the contact interactions in Eq.
(LABEL:c1) are SM gauge symmetric.

The four composite fields $F$: $U^{0}_{R}$, $U_{R}^{-}$, $D_{R}^{0}$,
$D^{-}_{R}$ are two-component right-handed Wely fermions. (To recall: N
(neutrino) corresponds to U (up-quark). E (electron) corresponds to D (down-
quark).

They form two right-handed SU(2) doublets: (i)
$U_{R}=(U^{0}_{R},U^{-}_{R})=\Pi^{a\dagger}_{7/6}u_{Ra}$ of u channel
hypercharge -7/6 +2/3= -1/2, (ii)
$D_{R}=(D^{0}_{R},D^{-}_{R})=\Pi^{a\dagger}_{1/6}d_{Ra}$ of d channel
hypercharge -1/6 -1/3= -1/2, where the SU(2) LQ composite boson doublets:
$\Pi^{a\dagger}_{1/6}=(\Pi^{-2/3}_{u_{a}},\Pi^{1/3}_{a})$ of u channel the
hypercharge -1/6, and
$\Pi^{a\dagger}_{7/6}=(\Pi^{-5/3}_{a},\Pi^{-2/3}_{d_{a}})$ of d channel the
hypercharge -7/6.

The SU(2) doublet massive Dirac fermions (four spinor components) are: (i)
$U_{D}=(U^{0}_{R}+\nu^{e}_{L},U^{-}_{R}+e_{L})$ of hypercharge -1/2 and Dirac
mass term $M_{F}\bar{U}_{D}U_{D}$; (ii)
$D_{D}=(D^{0}_{R}+\nu^{e}_{L},D^{-}_{R}+e_{L})$ of hypercharge -1/2 and Dirac
mass term $M_{F}\bar{D}_{D}D_{D}$.

There are two left-handed SU(2) singlets: (i)
$U^{s}_{La}=(\nu^{e}_{L},e_{L})\Pi^{a}_{1/6}$ of u channel of hypercharge
1/6-1/2=-1/3, (ii) $D^{s}_{La}=(\nu^{e}_{L},e_{L})\Pi^{a}_{7/6}$ of d channel
of hypercharge 7/6-1/2=2/3, where the SU(2) LQ composite boson doublets:
$\Pi^{a}_{1/6}=(\Pi^{2/3}_{u_{a}},\Pi^{-1/3}_{a})$ of u channel the
hypercharge 1/6, and $\Pi^{a}_{7/6}=(\Pi^{5/3}_{a},\Pi^{2/3}_{d_{a}})$ of d
channel the hypercharge 7/6.

The SU(2) singlet massive Dirac fermions (four spinor components) are: (i)
$U^{s}_{Da}=(U^{s}_{La}+d_{Ra})$ of hypercharge -1/3, charge e=Y=-1/3 and
Dirac mass term $M_{F}\bar{U}^{s}_{Da}U^{s}_{Da}$; (ii)
$D^{s}_{Da}=(D^{s}_{La}+u_{Ra})$ of hypercharge 2/3, charge e=Y=2/3 and Dirac
mass term $M_{F}\bar{D}^{s}_{Da}D^{s}_{Da}$.

Figure 2: A lepton $\ell$, two quarks $q$ ($u$-type) and $\bar{q}$ ($d$-type)
form a composite fermion $F$ via the contact interaction (dark blob)
$P_{L,R}(g_{*}^{2}/\Lambda^{2})$, where $P_{L,R}=(1\mp\gamma_{5})/2$. The thin
solid line represents an SM elementary fermion, and the thick double line
represents a composite fermion $F$. By a crossing symmetry applied to the
lepton line $\ell\rightarrow\ell^{\dagger}$ (dashed line) the same diagram
describes a $2\to 2$ production process $q\bar{q}\to\ell^{\dagger}F$.  Figure
3: We show the Feynman diagrammatic representation for the contact interaction
between the composite boson and its constituent quarks, where the thin solid
line represents an SM elementary fermion, double wave line represents a
composite boson, and the blob represents an interacting vertex
$(F_{\Pi}/\Lambda)^{2}P_{L,R}$. Figure 4: We show the Feynman diagrammatic
representation for the contact interaction between the composite fermion and
boson, where the thin solid line represents a SM elementary fermion, the
double solid line is a composite fermion and the double wave line represents a
composite boson and the blob represents an interacting vertex
$(F_{\Pi}/\Lambda)^{2}P_{L,R}$.

##  IX contact interaction of composite fermions

The contact interactions for the production and decay of a composite fermions
$F$ are:

$\displaystyle{\mathcal{L}}^{F}_{\rm
CI}={\mathcal{V}}_{F}+{\mathcal{V}}_{F}^{\dagger}.$ (24)

In the case of Eq. (3) and Table 8,

$\displaystyle{\mathcal{V}}_{\bar{U}^{0}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{U}^{0}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\nu^{e}_{L}),\quad
pp~{}\rightarrow\bar{U}^{0}_{L}\nu^{e}_{L};\quad
p\nu_{L}~{}\rightarrow\bar{U}^{0}_{L}u^{a}_{R},$ (25)
$\displaystyle{\mathcal{V}}_{\bar{U}^{+}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{U}^{+}_{L}u^{a}_{R})(\bar{u}^{a}_{R}e_{L}),\quad
pp~{}\rightarrow\bar{U}^{+}_{L}e_{L};\quad
ep~{}\rightarrow\bar{U}^{+}_{L}u^{a}_{R},$ (26)
$\displaystyle{\mathcal{V}}_{\bar{D}^{+}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{D}^{+}_{L}d^{a}_{R})(\bar{d}^{a}_{R}e_{L}),\quad
pp~{}\rightarrow\bar{D}^{+}_{L}e_{L};\quad
ep~{}\rightarrow\bar{D}^{+}_{L}d^{a}_{R},$ (27)
$\displaystyle{\mathcal{V}}_{\bar{D}^{0}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{D}^{0}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\nu^{e}_{L}),\quad
pp\rightarrow\bar{D}^{0}_{L}\nu^{e}_{R};\quad
p\nu_{L}\rightarrow\bar{D}^{0}_{L}d^{a}_{R},$ (28)

and their Hermitian conjugates,

$\displaystyle{\mathcal{V}}^{\dagger}_{U^{0}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{u}^{a}_{L}U^{0}_{R})(\bar{\nu}^{e}_{R}u_{La}),\quad
U^{0}_{R}\rightarrow\bar{u}^{a}_{L}(\bar{\nu}^{e}_{R}u_{aL}),$ (29)
$\displaystyle{\mathcal{V}}^{\dagger}_{U^{-}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{u}^{a}_{L}U^{-}_{R})(\bar{e}_{R}u^{a}_{L}),\quad
U^{-}_{R}\rightarrow\bar{u}^{a}_{L}(\bar{e}_{R}u_{aL}),$ (30)
$\displaystyle{\mathcal{V}}^{\dagger}_{D^{-}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{d}^{a}_{L}D^{-}_{R})(\bar{e}_{R}d^{a}_{L}),\quad
D^{-}_{R}\rightarrow\bar{d}^{a}_{L}(\bar{e}_{R}d_{La}),$ (31)
$\displaystyle{\mathcal{V}}^{\dagger}_{D^{0}}$ $\displaystyle=$
$\displaystyle\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{d}^{a}_{L}D^{0}_{R})(\bar{\nu}^{e}_{R}d^{a}_{L}),\quad
D^{0}_{R}\rightarrow\bar{d}^{a}_{L}(\bar{\nu}^{e}_{R}d_{La}).$ (32)

In the case of Eq. (4) and Table 6,

$\displaystyle{\mathcal{V}}_{\bar{E}^{-2/3}_{Lb}}=\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{E}^{-2/3}_{Lb}e_{R})(\bar{e}_{R}u_{Lb});\\!\\!\\!\quad\\!\\!\\!ee~{}\rightarrow\bar{E}^{-2/3}_{La}u_{Rb};\\!\\!\\!\quad\\!\\!\\!ep~{}\rightarrow\bar{E}^{-2/3}_{La}e_{R};$
(33)
$\displaystyle{\mathcal{V}}_{\bar{N}^{1/3}_{Lb}}=\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{N}^{1/3}_{Lb}\nu^{e}_{R})(\bar{\nu}^{e}_{R}d_{Lb});\quad\bar{\nu}_{R}\nu_{R}~{}\rightarrow\bar{N}^{1/3}_{La}d_{Ra};\quad
p\nu_{R}~{}\rightarrow\bar{N}^{1/3}_{La}\nu^{e}_{R},$ (34)
$\displaystyle{\mathcal{V}}_{\bar{E}^{1/3}_{Lb}}=\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{E}^{1/3}_{Lb}e_{R})(\bar{e}_{R}d_{Lb});\quad
ee~{}\rightarrow\bar{E}^{1/3}_{Lb}d_{Lb};\quad
ep~{}\rightarrow\bar{E}^{1/3}_{Lb}e_{R},$ (35)
$\displaystyle{\mathcal{V}}_{\bar{N}^{-2/3}_{Lb}}=\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{N}^{-2/3}_{Lb}\nu^{e}_{R})(\bar{\nu}^{e}_{R}u_{Lb});\\!\\!\\!\quad\\!\\!\\!\bar{\nu}_{R}\nu_{R}~{}\rightarrow\bar{N}^{-2/3}_{Lb}u_{Rb};\\!\\!\\!\quad\\!\\!\\!\bar{\nu}_{R}p~{}\rightarrow\bar{N}^{-2/3}_{Lb}\nu^{e}_{R},$
(36)

and their Hermitian conjugates,

$\displaystyle{\mathcal{V}}^{\dagger}_{E^{2/3}_{Rb}}\\!=\\!\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{e}_{L}E^{2/3}_{Rb})(\bar{u}^{b}_{R}e_{L});\\!\\!\quad\\!\\!E^{2/3}_{Rb}\rightarrow\bar{e}_{L}(\bar{u}^{b}_{R}e_{L}),$
(37)
$\displaystyle{\mathcal{V}}^{\dagger}_{N^{-1/3}_{Rb}}\\!=\\!\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{\nu}^{e}_{L}N^{-1/3}_{Rb})(\bar{d}^{b}_{R}\nu^{e}_{L});\\!\\!\\!\quad\\!\\!\\!N^{-1/3}_{Rb}\\!\\!\rightarrow\\!\bar{\nu}^{e}_{L}(\bar{d}^{b}_{R}\nu^{e}_{L}),$
(38)
$\displaystyle{\mathcal{V}}^{\dagger}_{E^{-1/3}_{Rb}}\\!=\\!\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{e}_{L}E^{-1/3}_{Rb})(\bar{d}^{b}_{R}e_{L});\\!\\!\\!\quad\\!\\!\\!E^{-1/3}_{Rb}\\!\\!\rightarrow\\!\bar{e}_{L}(\bar{d}^{b}_{R}e_{L}),$
(39)
$\displaystyle{\mathcal{V}}^{\dagger}_{N^{2/3}_{Rb}}\\!=\\!\frac{g_{*}^{2}}{\Lambda^{2}}(\bar{\nu}^{e}_{L}N^{2/3}_{Rb})(\bar{u}^{b}_{R}\nu^{e}_{L});\\!\\!\quad\\!\\!N^{2/3}_{Rb}\rightarrow\bar{\nu}^{e}_{L}(\bar{u}^{b}_{R}\nu^{e}_{L}).$
(40)

These are relevant contact interactions for phenomenological studies of
possible inelastic channels of composite-fermion production and decay in $pp$
or $ep$ collisions.

###  IX.1 Contact interaction of $\Pi^{0,\pm}$ composite boson and gauge
bosons

However, we shall consider other decay channels at the next leading order,
such as neutral composite boson decay to two SM gauge bosons
$\Pi^{0}_{u,d}\rightarrow\tilde{G}+\tilde{G}^{\prime}$. We need a figure for
$\Pi\rightarrow\tilde{G}+\tilde{G}^{\prime}$. Note that the discussions here
in this section do not apply to LQ composite bosons $\Pi^{Q}_{a}$ at this
leading order, because that SM gauge bosons are zero or integer electric
charges and the fractionally charged LQ composite bosons cannot decay to SM
gauge bosons.  The composite bosons $\Pi^{0,\pm}$ have two components: the
pesudo scalar and scalar component. As example
$\Pi^{+}=(g^{*}/F_{\Pi})^{2}(\bar{d}^{a}_{R}u_{La})$ (7) and
$\Pi^{0}_{u}=(g^{*}/F_{\Pi})^{2}(\bar{u}^{a}_{R}u_{La})$ (10) are [32]

$\displaystyle\Pi^{+}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}\bar{d}^{a}\frac{(1-\gamma_{5})}{2}u_{a}=(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}u_{a}-\bar{d}^{a}\gamma_{5}u_{a}\Big{)}$
(41) $\displaystyle\Pi^{0}_{u}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}\bar{u}^{a}\frac{(1-\gamma_{5})}{2}u_{a}=(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{u}^{a}u_{a}-\bar{u}^{a}\gamma_{5}u_{a}\Big{)},$
(42) $\displaystyle\Pi^{0}_{d}$ $\displaystyle=$
$\displaystyle(g^{*}/F_{\Pi})^{2}\bar{d}^{a}\frac{(1-\gamma_{5})}{2}d_{a}=(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}d_{a}-\bar{d}^{a}\gamma_{5}d_{a}\Big{)},$
(43)

which can be generalized to the 2nd and 3rd generations. In the following
studies of composite bosons decaying to two SM gauge bosons via a fermion
loop, for the sake of notations, we use the $\Pi^{0,\pm}$ also for the
pseudoscalar component with $\gamma_{5}$ presence, and introduce Higg-like
${\rm H}^{0,\pm}$ for scalar component without $\gamma_{5}$ presence. Namely,
henceforth, for pseudoscalar composite bosons

$\displaystyle\Pi^{+}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}\gamma_{5}u_{a}\Big{)}$
(44) $\displaystyle\Pi^{0}_{u}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{u}^{a}\gamma_{5}u_{a}\Big{)},$
(45) $\displaystyle\Pi^{0}_{d}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}\gamma_{5}d_{a}\Big{)},$
(46)

and for scalar Higg-like ${\rm H}^{0,\pm}$ bosons

$\displaystyle{\rm H}^{+}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}u_{a}\Big{)}$ (47)
$\displaystyle{\rm H}^{0}_{u}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{u}^{a}u_{a}\Big{)},$ (48)
$\displaystyle{\rm H}^{0}_{d}$ $\displaystyle=$
$\displaystyle(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{d}^{a}d_{a}\Big{)}.$ (49)

These can be generalized to the 2nd and 3rd families, ${\rm
H}^{0}_{t}=(1/2)(g^{*}/F_{\Pi})^{2}\Big{(}\bar{t}^{a}t_{a}\Big{)}$ can be the
candidate of the second Higgs particle, a massive and tightly bound state.

Analogously to $\pi^{0}\rightarrow\gamma\gamma$, the massive $\Pi^{0}_{u,d}$
composite boson (pesudo scalar component) can also decay into two gauge bosons
[32] :

$\displaystyle\Pi^{0}_{u,d}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+\gamma,$ (50) $\displaystyle\Pi^{0}_{u,d}$
$\displaystyle\rightarrow$ $\displaystyle\gamma+Z^{0},$ (51)
$\displaystyle\Pi^{0}_{u,d}$ $\displaystyle\rightarrow$ $\displaystyle
Z^{0}+Z^{0},$ (52) $\displaystyle\Pi^{0}_{u,d}$ $\displaystyle\rightarrow$
$\displaystyle W^{+}+W^{-},$ (53)

via the contact interaction

$\displaystyle{\mathcal{L}}^{\Pi^{0}}_{\tilde{G}\tilde{G}^{\prime}}=\sum_{i=u,d}\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu})\Pi^{0}_{i},$
(54)

where $g$ and $g^{\prime}$ represent the couplings of gauge bosons $A^{\mu}$
and $A^{\prime\nu}$ to the SM quarks $u$ and $d$ with different
$SU_{L}(2)$-isospin $i=u,d$. Actually, this effective contact interaction (54)
is an axial anomaly vertex, as a result of a triangle quark loop and standard
renormalization procedure in SM. The color factor $N_{c}=3$ comes from the
trace over color space of quarks in loop. The contact interaction (54) can be
written as

$\displaystyle{\mathcal{L}}^{\Pi^{0}}_{\tilde{G}\tilde{G}^{\prime}}=\sum_{i=u,d}\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}\frac{1}{4}(F^{\rho\mu})(F^{\prime\sigma\nu})\Pi^{0}_{i}.$
(55)

These amplitudes or effective interacting Lagrangians are the same,
independent of quark types and masses in triangle loop. Therefore, it is the
same and generalized for all generations. In addition, $\Pi^{0}_{u,d}$ decay
to two gluons

$\displaystyle\Pi^{0}_{u,d}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{g}+\tilde{g},$ (56)

via the contact interaction

$\displaystyle{\mathcal{L}}^{\Pi^{0}}_{\tilde{g}\tilde{g}^{\prime}}$
$\displaystyle=$
$\displaystyle\sum_{i=u,d}\frac{g_{s}^{2}/2}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}(\partial^{\rho}A^{\mu}_{a})(\partial^{\sigma}A^{\prime\nu}_{a})\Pi^{0}_{i}$
(57) $\displaystyle=$
$\displaystyle\sum_{i=u,d}\frac{g_{s}^{2}/2}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}\frac{1}{4}(G^{\rho\mu}_{a})(G^{\prime\sigma\nu}_{a})\Pi^{0}_{i},$

where the strong coupling $g_{s}$ and $\alpha_{s}=g_{s}^{2}/(4\pi)$, the trace
over color space gives $\hbox{tr}(T^{a}T^{b})=\delta^{ab}/2$, leading to
$(G^{\rho\mu}_{a})(G^{\prime\sigma\nu}_{a})$ summed over color index $a$, see
Ref. [37]. Here we note that the normal $\pi^{0}$ meson
$\pi^{0}=\bar{u}\gamma_{5}u-\bar{d}\gamma_{5}d$ in the $SU(3)$ quark model,
the $u$ and $d$ color charges are the same. Thus their triangle diagram
contributions to the anomaly, i.e., $\pi^{0}$ decays to two gluons, are the
same, but opposite in sign, therefore exactly canceled each other. As a
result, the total amplitude of the normal $\pi^{0}$ decay to two gluons is
zero. This is not the case for its decay to two photons, since the $u$ and $d$
have different electric charges. In our case, $\Pi^{0}_{u}$ and $\Pi^{0}_{d}$
are different states, and both decay to two gluons and amplitudes are the
same.

In addition, following the discussions and calculations of the neutral massive
$\Pi^{0}_{u,d}$ composite boson (pesudo scalar component) decay into two gauge
bosons (50-53) in Ref. [32], we can show the charged massive composite boson
$\Pi^{+}(u\bar{d})$ and $\Pi^{-}(d\bar{u})$ can decay. The neutral channels
are

$\displaystyle\Pi^{\pm}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+W^{\pm},$ (58) $\displaystyle\Pi^{\pm}$
$\displaystyle\rightarrow$ $\displaystyle Z^{0}+W^{\pm},$ (59)

via the contact interaction

$\displaystyle{\mathcal{L}}^{\Pi^{\pm}}_{\tilde{G}\tilde{G}^{\prime}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu})\Pi^{\pm}.$
(60)

The reason is that $\Pi^{\pm}$ masses are larger than the SM gauge bosons
$W^{\pm}$ and $Z^{0}$. There are two possibilities in Eq. (60). (i)
$\Pi^{+}AW^{+}$ interaction comes from the triangle loop made by quarks $uud$:

$\displaystyle{\mathcal{L}}^{\Pi^{+}}_{\tilde{\gamma}\tilde{W}^{+}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu})\Pi^{+}+h.c.,$
(61)

where fractional charge $g=(2/3)e$ and $SU_{L}(2)$ coupling
$g^{\prime}=g_{2}/(2\sqrt{2})$, and $h.c.$ represents $\Pi^{-}$ channel. (ii)
$\Pi^{+}AW^{+}$ interaction comes from the triangle loop made by quarks $udd$:

$\displaystyle{\mathcal{L}}^{\Pi^{+}}_{\tilde{\gamma}\tilde{W}^{+}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}\epsilon_{\mu\nu\rho\sigma}(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu})\Pi^{+}+h.c.,$
(62)

where fractional charge $g=-(1/3)e$ and $SU_{L}(2)$ coupling
$g^{\prime}=g_{2}/(2\sqrt{2})$, and $h.c.$ represents $\Pi^{-}$ channel. It is
the same for the $Z^{0}$ boson channel (59), replacing fractional charge $g$
by neutral gauge coupling $Z^{0}\bar{u}u$ or $Z^{0}\bar{d}d$.

###  IX.2 Contact interaction of Higgs-like ${\rm H}^{0,\pm}$ composite boson
and gauge bosons

Analogously, we also consider the channels of Higgs-like composite bosons
decay into two gauge bosons:

$\displaystyle{\rm H}^{0}_{u,d}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+\gamma,$ (63) $\displaystyle{\rm H}^{0}_{u,d}$
$\displaystyle\rightarrow$ $\displaystyle\gamma+Z^{0},$ (64)
$\displaystyle{\rm H}^{0}_{u,d}$ $\displaystyle\rightarrow$ $\displaystyle
Z^{0}+Z^{0},$ (65) $\displaystyle{\rm H}^{0}_{u,d}$ $\displaystyle\rightarrow$
$\displaystyle W^{+}+W^{-},$ (66) $\displaystyle{\rm H}^{0}_{u,d}$
$\displaystyle\rightarrow$ $\displaystyle\tilde{g}+\tilde{g},$ (67)

For the electroweak decay channels (63-66), the contact interaction is

$\displaystyle{\mathcal{L}}^{{\rm H}^{0}}_{\tilde{G}\tilde{G}^{\prime}}$
$\displaystyle=$
$\displaystyle\sum_{i=u,d}\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}-g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu}){\rm
H}^{0}_{i}$ (68) $\displaystyle=$
$\displaystyle\sum_{i=u,d}\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu}){\rm
H}^{0}_{i}$

where the gauge invariant transverse condition $\partial_{\mu}A^{\mu}=0$ is
used. For the QCD decay channels (67) to two gluons, the contact interaction
is

$\displaystyle{\mathcal{L}}^{{\rm
H}^{0}}_{\tilde{g}\tilde{g}^{\prime}}=\sum_{i=u,d}\frac{g_{s}^{2}/2}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu}_{a})(\partial^{\sigma}A^{\prime\nu}_{a}){\rm
H}^{0}_{i},$ (69)

where strong coupling $\alpha_{s}=g_{s}^{2}/4\pi$ and the trace over color
space $\hbox{tr}(T^{a}T^{b})=\delta^{ab}/2$. We compare the pesudoscalar
$\Pi^{0}$ channel contributions (55,69) and scalar ${\rm H}^{0}$ channel
contributions (68,69), they are of the same order. Therefore, the total decay
rate of a composite boson should be the sum of both pesudoscalar and scalar
channels’ contributions. In the previous EPJC paper, there was a error (factor
$3^{2}$ larger) in the rate of $\Pi^{0}$ decay to $W,Z,\gamma$, and the scalar
channel ${\rm H}^{0}$ contributions have not been accounted. But these do not
qualitatively change the conclusion/statement of the article, since the
branching ratios of decaying to two gauge bosons channels are far smaller (a
few order of mangnitudes) than the that of decaying to quarks channels.
Moreover, we did mention the composite boson decay to two gluons and can fuse
to Higgs boson.

The charged scalar ${\rm H}^{\pm}$ decay channels are

$\displaystyle{\rm H}^{\pm}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+W^{\pm},$ (70) $\displaystyle{\rm H}^{\pm}$
$\displaystyle\rightarrow$ $\displaystyle Z^{0}+W^{\pm}.$ (71)

via the contact interaction

$\displaystyle{\mathcal{L}}^{{\rm
H}^{\pm}}_{\tilde{G}\tilde{G}^{\prime}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu}){\rm
H}^{\pm}.$ (72)

The reason is that ${\rm H}^{\pm}$ masses are larger than the SM gauge bosons
$W^{\pm}$ and $Z^{0}$. There are two possibilities in Eq. (72). (i) ${\rm
H}^{+}AW^{+}$ interaction comes from the triangle loop made by quarks $uud$:

$\displaystyle{\mathcal{L}}^{{\rm
H}^{+}}_{\tilde{\gamma}\tilde{W}^{+}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu}){\rm
H}^{+}+h.c.,$ (73)

where fractional charge $g=(2/3)e$ and $SU_{L}(2)$ coupling
$g^{\prime}=g_{2}/(2\sqrt{2})$, and $h.c.$ represents $\Pi^{-}$ channel. (ii)
${\rm H}^{+}AW^{+}$ interaction comes from the triangle loop made by quarks
$udd$:

$\displaystyle{\mathcal{L}}^{{\rm
H}^{+}}_{\tilde{\gamma}\tilde{W}^{+}}=\frac{gg^{\prime}N_{c}}{4\pi^{2}F_{\Pi}}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\sigma}g_{\nu\rho})(\partial^{\rho}A^{\mu})(\partial^{\sigma}A^{\prime\nu}){\rm
H}^{+}+h.c.,$ (74)

where fractional charge $g=-(1/3)e$ and $SU_{L}(2)$ coupling
$g^{\prime}=g_{2}/(2\sqrt{2})$, and $h.c.$ represents ${\rm H}^{-}$ channel.
It is the same for the $Z^{0}$ boson channel (71), replacing fractional charge
$g$ by neutral gauge coupling $Z^{0}\bar{u}u$ or $Z^{0}\bar{d}d$.

##  X Phenomenology of the composite bosons in $pp$ and $ep$ collisions

In this section we study the phenomenology of the composite fermions in $pp$
collisions. We first outline its production and decay mode and then calculate
its cross section and decay width. This study leads us to the discussion on
the search for $\Pi$ that will be discussed in the next section.

###  X.1 Production and decay of composite bosons $\Pi^{0,\pm}$

As already specified in Sec. VII, the composite bosons $\Pi$ can be

$\displaystyle\Pi^{+}~{}(\bar{d}u,\bar{c}s,\bar{b}t),$
$\displaystyle\Pi^{-}~{}(\bar{u}d,\bar{s}c,\bar{t}b),$
$\displaystyle\Pi^{0}_{u}~{}(\bar{u}u,\bar{s}s,\bar{t}t),$
$\displaystyle\Pi^{0}_{d}~{}(\bar{d}d,\bar{c}c,\bar{b}b),$ (75)

for the charge sector $Q=+,-,0$ respectively, and the corresponding composite
bosons of the higher generation of families in the SM.

The kinematics of the processes is derived in the center of mass frame of $pp$
collisions and virtual processes of $\Pi$ are not considered. If the energy
$\sqrt{s}$ in the parton centre of mass frame is larger than composite boson
masses, the following resonant process can occur:

$\displaystyle pp$ $\displaystyle\rightarrow$ $\displaystyle~{}\Pi$ (76)

where the composite bosons $\Pi$ (75) formed by Standard Model fermions of the
higher generation of families. The kinematics of final states is simple in the
centre of mass frame of $pp$ collisions.

If we neglect the quark-family mixing, the previous process can manifest at
parton level as:

$\displaystyle u~{}\bar{d}$ $\displaystyle\rightarrow$ $\displaystyle\Pi^{+},$
(77) $\displaystyle d~{}\bar{u}$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{-},$ (78) $\displaystyle u~{}\bar{u}$
$\displaystyle\rightarrow$ $\displaystyle\Pi^{0}_{u},$ (79) $\displaystyle
d~{}\bar{d}$ $\displaystyle\rightarrow$ $\displaystyle\Pi^{0}_{d}.$ (80)

The composite boson $\Pi$ can decay through the two different channels: two SM
quarks $qq^{\prime}$, via the contact interactions in Eqs. (16-18), two SM
gauge bosons $\tilde{G}\tilde{G}^{\prime}$ via the contact interactions in
Eqs. (50-56). Then the composite fermion $\Pi$ decays as:

$\displaystyle\Pi~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}qq^{\prime},$ (81) $\displaystyle\Pi~{}$
$\displaystyle\rightarrow$ $\displaystyle\tilde{G}\tilde{G}^{\prime}.$ (82)

The full decay chain is:

$\displaystyle pp$ $\displaystyle\rightarrow$
$\displaystyle\Pi\rightarrow~{}qq^{\prime},$ (83) $\displaystyle pp$
$\displaystyle\rightarrow$
$\displaystyle\Pi\rightarrow~{}\tilde{G}\tilde{G}^{\prime}.$ (84)

It appears clear, considering all the possible flavours of $\Pi$ (75), that a
large range of final states is possible. The cross section of the process
$pp\rightarrow\Pi$, the decay branching ratios of $\Pi$, and the final states
and their topologies are discussed below.

Figure 5: (Color Online) On the left panel, we show the production cross
section of $pp\rightarrow\Pi$ as a function of $m_{{}_{\Pi}}$ for the case
$g^{2}_{{}_{\rm Y}}=\left(\frac{F_{\pi}}{\Lambda}\right)=0.5$ and at a centre
of mass energy $\sqrt{s}=13$ TeV. The solid red line represents the results of
an analytical and numerical calculation based on Eqs. (109) and (118) and the
filled circles (black) represent the results from our implementation of the
model in CalcHEP. We find good agreement. On the right panel, we plot the
decay width of composite boson $\Pi$ as a function of its mass $m_{{}_{\Pi}}$
for the case $g^{2}_{{}_{\rm Y}}=\left(\frac{F_{\pi}}{\Lambda}\right)=0.5$.
Again, we observe a good agreement between the expectation from a CalcHEP
simulation and the analytical result based on Eq. (119). here 1. we need a
plot for the decay rate; 2. we need to think the value $g^{2}_{{}_{\rm
Y}}=\left(\frac{F_{\pi}}{\Lambda}\right)$, in connection with the previous
fermion paper.

###  X.2 Production and decay of quark-lepton composite bosons
$\Pi^{5/3,-1/3,2/3}_{a}$

As already specified in Sec. VII, the quark-lepton composite bosons $\Pi$ can
be

$\displaystyle\Pi^{5/3}_{a}~{}(\bar{e}u,\bar{\mu}c,\bar{\tau}t),$
$\displaystyle\Pi^{-1/3}_{a}~{}(\bar{\nu}^{e}d,\bar{\nu}^{\mu}s,\bar{\nu}^{\mu}b),$
$\displaystyle\Pi^{2/3}_{u_{a}}~{}(\bar{\nu}^{e}u,\bar{\nu}^{\mu}c,\bar{\nu}^{\tau}t),$
$\displaystyle\Pi^{2/3}_{d_{a}}~{}(\bar{e}d,\bar{\mu}s,\bar{\tau}b),$ (85)

for the charge sector $Q=+5/3,-1/3,2/3$ respectively, and the corresponding
composite bosons of the higher generation of families in the SM.

The kinematics of the processes is derived in the center of mass frame of $pp$
collisions and virtual processes of $\Pi$ are not considered. If the energy
$\sqrt{s}$ in the parton centre of mass frame is larger than composite boson
masses, the following resonant process can occur:

$\displaystyle p~{}e$ $\displaystyle\rightarrow$ $\displaystyle~{}\Pi_{a}$
(86)

where the composite bosons $\Pi_{a}$ (75) formed by Standard Model fermions of
the higher generation of families. The kinematics of final states is simple in
the centre of mass frame of $ep$ collisions.

If we neglect the quark-family mixing, the previous process can manifest at
parton level as:

$\displaystyle u~{}\bar{e}$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{5/3}_{a},$ (87) $\displaystyle d~{}\bar{\nu}^{e}$
$\displaystyle\rightarrow$ $\displaystyle\Pi^{-1/3}_{a},$ (88) $\displaystyle
u~{}\bar{\nu}^{e}$ $\displaystyle\rightarrow$ $\displaystyle\Pi^{2/3}_{a},$
(89) $\displaystyle d~{}\bar{e}$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{2/3}_{a}.$ (90)

We find that the quark-lepton composite bosons $\Pi^{5/3}_{a}$ and
$\Pi^{2/3}_{a}$ can be produced by $ep$ collisions.

The composite boson $\Pi_{a}$ can decay through the different channels: SM
quark $q$ and lepton $\ell$, via the contact interactions in Eqs. (19-21).
Then the composite fermion $\Pi$ decays as:

$\displaystyle\Pi_{a}~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}\ell~{}q,$ (91)

the branching ratio of this decay is the approximately same as the decay of
the $\Pi^{0,\pm}(\bar{q}q)$, which has been calculated. The full decay chain
is:

$\displaystyle e~{}p$ $\displaystyle\rightarrow$
$\displaystyle\Pi_{a}\rightarrow~{}\ell q.$ (92)

It appears clear, considering all the possible flavours of $\Pi_{a}$ (85),
that a large range of final states is possible. The cross section of the
process $ep\rightarrow\Pi_{a}$, the decay branching ratios of $\Pi_{a}$, and
the final states and their topologies are discussed below. Composite bosons
and quark-lepton composite bosons have same cross sections and decay rates, by
exchanging (15).

Note that Quark-lepton composite bosons do not possess contact interactions
with SM gauge bosons like (50-56).

###  X.3 Production and decay of quark-lepton composite bosons
$\Pi^{5/3,-1/3,2/3}_{a}$ in $pp$ collisions

As shown in Figures 6 and 7, the lepton-quark state ca be also produced by
gluon and quark scattering

$\displaystyle g+q\rightarrow\ell+\Pi_{a},$ (93)

where $\Pi_{a}=(g^{*}/F_{\pi})^{2}(q_{a}\ell)$, see table 2, and the
interaction strength is $g_{{}_{Y}}$, see integrating Lagrangian Eqs. (16-22)
for the $\Pi q\ell$ vertex. The gluon and single quark interacting interacting
Lagranian is the normal QCD one, with the strength $\sqrt{\alpha_{s}}=g_{3}$.
This is the same for all three generations as show in Fig. 6.

For the reasons that composite bosons and fermions carry the quantum numbers
of SM $SU_{c}(3)\times SU_{L}(2)\times U_{Y}(1)$, they gauge couplings
$g_{3},g_{2},g_{1}$ to SM gauge bosons are well defined [32]. For the
$SU_{c}(3)$ sector,
$[(\partial_{\mu}+\cdot\cdot\cdot+ig_{3}T^{a}A_{\mu}^{a})\Pi^{Q}]^{2}$ gives
the interaction $g_{3}g_{3}\Pi_{a}^{Q}(\Pi_{a}^{Q})^{\dagger}$ of two gluons
and two leptoquark composite bosons ($[T^{a},T^{b}]=if^{abc}T^{c}$ and
$\hbox{tr}(T^{a}T^{b})=\delta^{ab}/2$),

$\displaystyle
g^{2}_{3}T^{a}T^{b}A^{a}_{\mu}A^{b}_{\mu}\Pi^{Q}(\Pi^{Q})^{\dagger},$ (94)

where we drop the color index $a$ of leptoquark boson $\Pi_{a}^{Q}$ (table 2),
for the process

$\displaystyle g+g~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}\Pi^{Q}+(\Pi^{Q})^{\dagger},$ (95)

as shown in Francesco slide 7. This is the gluon-gluon collision producing two
leptoquark bosons, kinematically, CM energy of two gluons should be larger
than $2M_{\Pi}$.

Analogously, the gluon and composite fermion interacting Lagrangian is from
$\bar{F}^{Q}_{L}(\partial_{\mu}+\cdot\cdot\cdot+ig_{3}T^{a}\gamma_{\mu}A_{\mu}^{a})F^{Q}_{L}$
gives the interaction $g\bar{F}^{Q}F^{Q}$ of gluon and composite fermion

$\displaystyle+ig_{3}T^{a}A^{a}_{\mu}\bar{F}^{Q}_{R}\gamma^{\mu}F^{Q}_{R}+ig_{3}T^{a}A^{a}_{\mu}\bar{F}^{Q}_{L}\gamma^{\mu}F^{Q}_{L},$
(96)

where the composite fermion $F^{Q}_{R}$ is $E_{R},N_{R},U_{R}$ or $D_{R}$ and
their Hermitian conjugates of left-handed counterpart $F^{Q}_{L}$ that are
bound states of a quark and composite boson $\Pi$, see the Table I and II of
our previous article. This is the case for the composite Fermion made by three
quarks. It needs study (table) for the cases of the composite quarklepto
Fermion states made by one lepton and two quarks or two leptons and one quark.
This interaction (96) accounts for the gluon-gluon fusing to composite fermion
pair

$\displaystyle g+g~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}~{}F+\bar{F},$ (97)

The amplitude needs to be calculated, see Fig. 6. Answering Francesco
question: the composite fermion $F$ can be a bound state of a fermion $f$ and
a leptoquark $\Pi_{a}$, $F$ decays to $f$ and $\Pi_{a}$ … This needs to study
in details.

We consider the processes of $q\bar{q}$ collisions to produce leptoquark bosob
states,

$\displaystyle q\bar{q}\rightarrow\Pi+\Pi^{\dagger},$ (98)

where $\Pi$ stands for leptoquark states listed in Table 2. From Eqs.
(16)-(22), we can obtain the effective contact Lagrangians

$\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{d}^{a}_{R}u_{La})(\Pi^{2/3}_{u_{a}})^{\dagger}\Pi^{-1/3}_{a}+{\rm
h.c.},$ (99) $\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{d}^{a}_{R}u_{La})(\Pi^{2/3}_{d_{a}})^{\dagger}(\Pi^{5/3}_{a})+{\rm
h.c.},$ (100) $\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{d}^{a}_{R}d_{La})(\Pi^{-1/3}_{a})^{\dagger}\Pi^{-1/3}_{a}+{\rm
h.c.},$ (101) $\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{d}^{a}_{R}d_{La})(\Pi^{2/3}_{d_{a}})^{\dagger}\Pi^{2/3}_{d_{a}}+{\rm
h.c.},$ (102) $\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{u}^{a}_{R}u_{La})(\Pi^{5/3}_{a})^{\dagger}\Pi^{5/3}_{a}+{\rm
h.c.},$ (103) $\displaystyle\frac{g^{2}_{{}_{\rm
Y}}}{M_{\Pi}}(\bar{u}^{a}_{R}u_{La})(\Pi^{2/3}_{u_{a}})^{\dagger}\Pi^{2/3}_{u_{a}}+{\rm
h.c.}.$ (104)

Compared with Eqs. (16)-(22) with one leptoquark boson state, we have one more
leptoquark Yukawa coupling $g_{{}_{Y}}\sin{\mathcal{O}}(1)$ comes from the two
form factors for two composite leptoquark boson states.

Another possible effective contact Lagrangian comes comes from bar q q gluon
slef-energy diagram that couples two $\Pi$ and $\Pi$

$\displaystyle\frac{\alpha_{s}^{2}}{F_{\pi}}(\bar{q}q)(\Pi^{Q}_{a})^{\dagger}\Pi_{a}^{Q}$

to be checked, and see what others have done something similarly.

Figure 6: We show the Feynman diagrammatic representation for the contact
interaction between the composite boson and its constituent quarks, where the
thin solid line represents an SM elementary fermion (quark and lepton), double
wave line represents a composite boson, and the blob represents an interacting
vertex $g_{{}_{Y}}\sim{\mathcal{O}}(1)$, see Eqs. (16-22). Figure 7: Francesco
slide 1 Figure 8: Francesco slide 4. all are possible in our model, replacing
Z’ or W’ by a proper Pi composite boson, and interacting vertex is given by
Eqs. (16-22) for the leptoquark $\Pi q\ell$ vertex, they are the same as those
for normal qq composite, given in our previous article.

We turn to the pair production of composite particles (fermion and boson) by
SM gauge bosons, in particularly by gluons. The composite particles are made
of SM quarks and leptons. Therefore, they have SM quantum numbers to couple to
SM gauge bosons. These couplings in principle are new parameters of the
theory. However, If SM gauge bosons, here we particularly mean gluons, have
energies comparable to composite scale, namely the composite particle masses
$M_{F,\Pi}\sim{\mathcal{O}}(TeV)$, then gluons interact with the quark
constitute inside the composite particle. In this case, the strength of gluon
coupling to composite particle can be approximately represented by QCD strong
coupling $g_{s}=\sqrt{\alpha_{s}}$. This is the case for the pair productions
of composite particle $\Pi,F$ and anti composite particle
$\Pi^{\dagger},F^{\dagger}$ by gluon and gluon collisions,

$\displaystyle g+g~{}$ $\displaystyle\rightarrow$
$\displaystyle~{}\Pi+\Pi^{\dagger}~{}~{}{\rm or}~{}~{}F+F^{\dagger},$ (105)

The reason is that in center of mass frame, gluon and gluon collision energy
must be larger than the energy threshold $2M_{F,\Pi}$ to produce two composte
particles flying apart, back-to-back. The composite particle and anti
composite particle have the same mass. Here, we may consider the lepton-quark
composite states, since their decay modes (topologies) are distinct from the
SM QCD background. But the discussions are the same for the pair productions
of other composite particles in general.

This situation is very much like the Breit-Wheeler process of electron-
positron pair production by light-light collision $\gamma+\gamma\rightarrow
e^{+}+e^{-}$. Thus, learning from the QED calculation as shown two tree-level
Feynman diagrams in Fig. 9, we can write the cross-section as

$\displaystyle\sigma_{gg\rightarrow FF^{\dagger}~{}{\rm
or}~{}\Pi\Pi^{\dagger}}~{}$ $\displaystyle\approx$
$\displaystyle~{}\frac{\pi}{2}\left(\frac{\alpha_{s}}{M_{F,\Pi}}\right)^{2}(1-\hat{\beta}^{2})\Big{[}2\hat{\beta}^{2}(\hat{\beta}^{2}-2)+(3-\hat{\beta}^{4})\ln\Big{(}\frac{1+\hat{\beta}^{2}}{1-\hat{\beta}^{2}}\Big{)}\Big{]},\quad\hat{\beta}^{2}=\frac{|{\bf
p}|}{{\mathcal{E}}_{gg}}$ (106)

where ${\mathcal{E}}_{gg}$ is the two gluons colision energy in CM. Here we
use approximation $\sigma_{gg\rightarrow FF^{\dagger}~{}{\rm
or}~{}\Pi\Pi^{\dagger}}~{}\approx...$, because we approximate the coupling
between gluon and composite particle to be the QCD coupling $g_{s}$. For the
energy ${\mathcal{E}}_{gg}\gg M_{F,\Pi}$, the pair-production cross-section
yields

$\displaystyle\sigma_{gg\rightarrow FF^{\dagger}~{}{\rm
or}~{}\Pi\Pi^{\dagger}}~{}$ $\displaystyle\approx$
$\displaystyle~{}\pi\left(\frac{\alpha_{s}}{M_{F,\Pi}}\right)^{2}\left(\frac{M_{F,\Pi}}{{\mathcal{E}}_{gg}}\right)^{2}.$
(107)

Only the composite particle masses are involved, no other new parameters like
$g_{{}_{Y}}$. This process should have the biggest cross section (absorption
of gluons), if the gluons collision energy is over the energy threshold
$M_{F,\Pi}$ of pair production of composite particles. The decay modes of
back-to-back composite particles follow discussions before.

Figure 9: We show the Feynman diagrammatic representation for the pair
production of composite partilces by gluon-gluon collisions. The gluon
coupling to lepton-quark composite particles is approximated by the gluon QCD
coupling $g_{s},\alpha_{s}=g_{s}^{2}$ to a quark.

###  X.4 Vector-boson scatterings in LHC

The massive vector bosons for instance $W^{+}W^{-}$ radiated from two quark
lines can scatter and form the intermediate state $\Pi^{0}$, see Eqs. (53-52)
and (58-59-59). These should give contributions to the Vector-boson
scatterings (VBS) discussed in the articles linked below. The final states
topology are two quark-jets and massive vector-bosons non-lepton, full-lepton,
semi-lepton decays. In addition, the peculiar topology in final states we have
from Eqs. (50)-(53) is

$\displaystyle W^{+}W^{-}\rightarrow\Pi^{0}\rightarrow\gamma\gamma,$ (108)

namely two jets and two photons with larger branching ratio. In the SM, the
massive vector-boson scatterings does not involve photons.

https://arxiv.org/pdf/1708.02812.pdf

https://arxiv.org/pdf/2102.10991.pdf

https://www.sciencedirect.com/science/article/pii/S2405428318300789

$https://indico.cern.ch/event/253774/contributions/567168/attachments/443081/614598/talk_{c}eballos_{v}bs.pdf$

###  X.5 Cross sections, decay widths and branching ratios

####  X.5.1 Cross sections

The partonic cross section of $qq^{\prime}\rightarrow\Pi$ is calculated by
standard methods via the contact interaction in Eqs. (16-18), check again this
equation

$\displaystyle\hat{\sigma}_{{}_{\Pi}}(\hat{s},m_{{}_{\Pi}})=\frac{\pi}{12}\left(\frac{F_{\Pi}}{\Lambda}\right)^{4}\delta(\hat{s}-m_{{}_{\Pi}}^{2}),$
(109)

where $\sqrt{\hat{s}}$ stands for the parton center-mass-energy of $pp$
collisions in LHC experiments. These are the same for all channels: Composite
bosons $\Pi^{0,\pm}$ (7-10) are produced in $pp$ collisions, then decay to two
jets,

$\displaystyle pp(u\bar{d})$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{+}\rightarrow{\rm two~{}jets}~{}(u\bar{d})$ (110)
$\displaystyle pp(d\bar{u})$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{-}\rightarrow{\rm two~{}jets}~{}(d\bar{u})$ (111)
$\displaystyle pp(u\bar{u})$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{0}_{u}\rightarrow{\rm two~{}jets}~{}(u\bar{u})$ (112)
$\displaystyle pp(d\bar{d})$ $\displaystyle\rightarrow$
$\displaystyle\Pi^{0}_{d}\rightarrow{\rm two~{}jets}~{}(d\bar{d})$ (113)

as illustrated by Feynman diagram Figure 3. The most favorable channels in
$pp$ collisions should be the productions of charged composite bosons
$\Pi^{\pm}$ (7). The final states are two back-to-back two standard jets
(dijet).

The cross section (109) should be also the same for all channels: quark-lepton
composite bosons $\Pi_{a}^{5/3,-1/3,2/3}$ (11-14) are produced in $ep$
collisions, then decay to a lepton and a jet,

$\displaystyle ep(u\bar{e})$ $\displaystyle\rightarrow$
$\displaystyle\Pi_{a}^{5/3}\rightarrow{\rm lepton+jet}~{}(u\bar{e})$ (114)
$\displaystyle ep(d\bar{\nu})$ $\displaystyle\rightarrow$
$\displaystyle\Pi_{a}^{-1/3}\rightarrow{\rm lepton+jet}~{}(d\bar{\nu})$ (115)
$\displaystyle ep(u\bar{\nu})$ $\displaystyle\rightarrow$
$\displaystyle\Pi_{u_{a}}^{2/3}\rightarrow{\rm lepton+jet}~{}(u\bar{\nu})$
(116) $\displaystyle ep(d\bar{e})$ $\displaystyle\rightarrow$
$\displaystyle\Pi_{d_{a}}^{2/3}\rightarrow{\rm lepton+jet}~{}(d\bar{e})$ (117)

as illustrated by Feynman diagram Figure 3. new figure The most favorable
channels in $ep$ collisions should be the productions of charged composite
bosons $\Pi_{a}^{5/3}$ (114) and $\Pi_{d_{a}}^{2/3}$ (117). The final states
are back-to-back a lepton and a standard jet.

We consider the production cross sections for the composite fermions $F$ in
$pp$ collisions expected at the LHC collider according to Feynman’s parton
model. The QCD factorization theorem allows to obtain any hadronic cross
section (e.g. in $pp$ collisions) in terms of a convolution of the hard
partonic cross sections $\hat{\sigma}$, evaluated at the parton centre of mass
energy $\sqrt{\hat{s}}=\sqrt{\tau s}$, with the universal parton distribution
functions $f_{a}(x,{\hat{\cal Q}})$ which depend on the parton longitudinal
momentum fractions $x$, and on the factorization scale $\hat{\cal Q}$: check
again this equation

$\displaystyle\sigma_{{}_{\Pi}}=\sum_{ij}\\!\\!\intop^{1}_{\frac{m_{{}_{\Pi}}^{2}}{s}}\\!\\!d\tau\\!\\!\intop_{\tau}^{1}\\!\\!\frac{dx}{x}f_{i}(x,\hat{\cal
Q}^{2})f_{j}(\frac{\tau}{x},\hat{\cal Q}^{2})\hat{\sigma}(\tau
s,m_{{}_{\Pi}})\,.$ (118)

The factorization and renormalization scale ${\cal Q}$ is generally fixed at
the value of the mass that is being produced. The parametrization of the
parton distribution function is NNPDF3.0 [39] and the factorization scale has
been chosen as $\hat{\cal Q}=m_{{}_{\Pi}}$. The results for the four composite
bosons at LHC are shown in figure (5), where we used $g_{{}_{\rm
Y}}=\left(\frac{F_{\Pi}}{\Lambda}\right)^{2}=0.5$.

The right panel of Fig. 5 shows the agreement between analytical calculations
based on Eqs. (109) and (118) for the case of the composite fermion $F$, and
the results of simulations with CalcHEP where the model with four-fermion
interactions has been implemented. We remark the quite good agreement that
validates our model implementation in CalcHEP.

####  X.5.2 Decay widths

The decay width of the $\Pi$ boson to two quarks is simply calculated by using
the effective contact Lagrangian in Eq. (16) and (17),

$\displaystyle\Gamma(\Pi\to
qq^{\prime})=\frac{3}{16\pi}\,\left(\frac{F_{\Pi}}{\Lambda}\right)^{4}\,m_{{}_{\Pi}}.$
(119)

For the case that $\Pi$ equals to $\Pi^{+}$ or $\Pi^{-}$ composite boson,
$\Pi\rightarrow qq^{\prime}$ of Eq. (119) is the only decay channel, see Eqs.
(7) and (16). The $\Pi^{0}_{u,d}$ composite bosons (pesudo scalar component),
instead, can also decay to two gauge bosons $\tilde{G}\tilde{G}^{\prime}$
(50-56), according to the contact interaction (54), the corresponding decay
widths are [32] :

$\displaystyle\Gamma_{\Pi^{0}_{u}\rightarrow\gamma\gamma}$ $\displaystyle=$
$\displaystyle\left(\frac{4}{9}\right)^{2}\Gamma,$ (120)
$\displaystyle\Gamma_{\Pi^{0}_{d}\rightarrow\gamma\gamma}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{9}\right)^{2}\Gamma,$ (121)
$\displaystyle\Gamma_{\Pi^{0}_{u}\rightarrow\gamma Z^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sin^{2}2\theta_{W}}\left(\frac{1}{3}-\frac{4}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma,$
(122) $\displaystyle\Gamma_{\Pi^{0}_{d}\rightarrow\gamma Z^{0}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sin^{2}2\theta_{W}}\left(\frac{1}{6}-\frac{1}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma,$
(123) $\displaystyle\Gamma_{\Pi^{0}_{u}\rightarrow Z^{0}Z^{0}}$
$\displaystyle\\!=\\!$
$\displaystyle\left(\frac{1/2\\!-\\!(2/3)\sin^{2}\theta_{W}}{\sin^{2}2\theta_{W}}\right)^{\\!2}\Gamma,$
(124) $\displaystyle\Gamma_{\Pi^{0}_{d}\rightarrow Z^{0}Z^{0}}$
$\displaystyle\\!=\\!$
$\displaystyle\left(\frac{-1/2\\!+\\!(1/3)\sin^{2}\theta_{W}}{\sin^{2}2\theta_{W}}\right)^{\\!2}\Gamma,$
(125) $\displaystyle\Gamma_{\Pi^{0}_{u,d}\rightarrow W^{+}W^{-}}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{8\sin^{2}\theta_{W}}\right)^{2}\Gamma,$ (126)

where $\theta_{W}$ is the Weinberg angle and fine structure constant
$\alpha=e^{2}\approx 1/137$,

$\Gamma=\left(\frac{\alpha N_{c}}{\pi
F_{\Pi}}\right)^{2}\frac{m_{\Pi^{0}_{u,d}}^{3}}{64\pi},$ (127)

and the number of colors $N_{c}=3$. Total decay rate $\Gamma^{\rm
tot}(\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime})$ is the sum over all
contributions from Eqs. (120-126). The total $\Pi^{0}_{u,d}$-decay rate reads

$\Gamma_{\rm tot}(\Pi^{0}_{u,d})=\Gamma(\Pi^{0}_{u,d}\rightarrow
qq^{\prime})+\Gamma^{\rm
tot}(\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime}),$ (128)

where $\Gamma(\Pi^{0}_{u,d}\rightarrow qq^{\prime})$ is given by Eq. (119).
Based on these results, we calculate the exact branching ratios of different
channels for different parameters of the model.

Following the calculations of $\Pi^{0}$ decaying to two gauge bosons
$GG^{\prime}$ [32], we can obtain the rates of charged composite bosons decay
to two gauge bosons (58) and (59):

$\displaystyle\Gamma_{\Pi^{+}\rightarrow\gamma W^{+}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{8\sin^{2}\theta_{W}}\right)\left(\frac{2}{3}\right)^{2}\Gamma,$
(129) $\displaystyle\Gamma_{\Pi^{-}\rightarrow\gamma W^{-}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{8\sin^{2}\theta_{W}}\right)\left(\frac{1}{3}\right)^{2}\Gamma,$
(130) $\displaystyle\Gamma_{\Pi^{+}\rightarrow W^{+}Z^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\sin^{2}\theta_{W}\sin^{2}2\theta_{W}}\left(\frac{1}{3}-\frac{4}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma,$
(131) $\displaystyle\Gamma_{\Pi^{-}\rightarrow W^{-}Z^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\sin^{2}\theta_{W}\sin^{2}2\theta_{W}}\left(-\frac{1}{6}-\frac{1}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma,$
(132) $\displaystyle\Gamma_{\Pi^{+}\rightarrow\gamma W^{+}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{8\sin^{2}\theta_{W}}\right)\Big{[}\left(\frac{2}{3}\right)^{2}+\left(\frac{1}{3}\right)^{2}\Big{]}\Gamma,$
(133) $\displaystyle\Gamma_{\Pi^{-}\rightarrow\gamma W^{-}}$ $\displaystyle=$
$\displaystyle\Gamma_{\Pi^{+}\rightarrow\gamma W^{+}},$ (134)
$\displaystyle\Gamma_{\Pi^{+}\rightarrow W^{+}Z^{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\sin^{2}\theta_{W}\sin^{2}2\theta_{W}}\left(\frac{1}{3}-\frac{4}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma$
(135) $\displaystyle+$
$\displaystyle\frac{1}{8\sin^{2}\theta_{W}\sin^{2}2\theta_{W}}\left(-\frac{1}{6}-\frac{1}{9}\sin^{2}\theta_{W}\right)^{2}\Gamma,,$
(136) $\displaystyle\Gamma_{\Pi^{-}\rightarrow W^{-}Z^{0}}$ $\displaystyle=$
$\displaystyle\Gamma_{\Pi^{+}\rightarrow W^{+}Z^{0}}.$ (137)

We have not considered these decay channels in previous article.

Following these discussions and using Eqs. (120-126), we can approximately
estimate the decay-rate ratios

$\displaystyle\Gamma_{\Pi^{0}\rightarrow\gamma
Z^{0}}/\Gamma_{\Pi^{0}\rightarrow\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{\left(9/10-\sin^{2}\theta_{W}\right)^{2}}{\sin^{2}2\theta_{W}}\approx
0.63,$ $\displaystyle\Gamma_{\Pi^{0}\rightarrow
W^{+}W^{-}}/\Gamma_{\Pi^{0}\rightarrow\gamma\gamma}$ $\displaystyle\approx$
$\displaystyle\left(\frac{9}{40\sin^{2}\theta_{W}}\right)^{2}\approx 0.96,$
$\displaystyle\Gamma_{\Pi^{0}\rightarrow
Z^{0}Z^{0}}/\Gamma_{\Pi^{0}\rightarrow\gamma\gamma}$ $\displaystyle=$
$\displaystyle\left(\frac{(9/10)-(9/5)\sin^{2}\theta_{W}+\sin^{4}\theta_{W}}{\sin^{2}2\theta_{W}}\right)^{2}\approx
0.58,$ (138)

where the value $\sin^{2}\theta_{W}\approx 0.23$ is approximately adopted to
obtain numbers.

These relations provide a possibility to verify the validity of such an
effective theory of composite particles, if the resonance of diphoton channel
and other diboson channels are also observed and measured in high-energy
experiments, for instance the on-going ATLAS and CMS experiments in the LHC
$pp$ collisions.

####  X.5.3 Branching ratios

The branching ratios of the charged composite bosons $\Pi^{\pm}$ decay to two
quarks $qq^{\prime}$,

$\displaystyle{\mathcal{B}}(\Pi^{\pm}\rightarrow
qq^{\prime})=\frac{\Gamma(\Pi^{\pm}\rightarrow qq^{\prime})}{\Gamma_{\rm
tot}(\Pi^{\pm})}\approx 1,$ (139)

The branching ratios of the $\Pi^{0}_{u,d}$ decay to two quarks $qq^{\prime}$,

$\displaystyle{\mathcal{B}}(\Pi^{0}_{u,d}\rightarrow
qq^{\prime})=\frac{\Gamma(\Pi^{0}_{u,d}\rightarrow qq^{\prime})}{\Gamma_{\rm
tot}(\Pi^{0}_{u,d})},$ (140)

and the $\Pi^{0}_{u,d}$ decay to two gauge bosons
$\tilde{G}\tilde{G}^{\prime}$,

$\displaystyle{\mathcal{B}}(\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime})=\frac{\Gamma^{\rm
tot}(\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime})}{\Gamma_{\rm
tot}(\Pi^{0}_{u,d})}.$ (141)

As a result, the cross sections of these processes are:

$\displaystyle\sigma(pp\rightarrow\Pi\rightarrow qq^{\prime})$
$\displaystyle=$ $\displaystyle\sigma(pp\rightarrow\Pi)$ (142)
$\displaystyle\times$
$\displaystyle\Big{[}{\mathcal{B}}(\Pi^{\pm}\rightarrow\bar{q}q^{\prime})+{\mathcal{B}}(\Pi^{0}_{u,d}\rightarrow\bar{q}q^{\prime})\Big{]},$

and

$\displaystyle\sigma(pp\rightarrow\Pi\rightarrow\tilde{G}\tilde{G}^{\prime})$
$\displaystyle=$
$\displaystyle\sigma(pp\rightarrow\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime})$
(143) $\displaystyle\times$
$\displaystyle{\mathcal{B}}(\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime}).$

which depend on composite boson mass $m_{{}_{\Pi}}$ and in particular on decay
constant $F_{\Pi}$. All channels of composite bosons $\Pi$ (75) production and
decay give the same results at this level of approximation by using contact
interactions only.

Figure 10:  (Color Online). The branching ratios of $\Pi\rightarrow
qq^{\prime}$ and $\Pi\rightarrow G\tilde{G}^{\prime}$ are plotted with respect
to the ratio $m_{{}_{\Pi}}/F_{\Pi}$ for the values of
$g^{2}_{{}_{Y}}=F_{\Pi}/\Lambda=0.5,0.1,0.02$ ???

###  X.6 Parameters are $m_{{}_{\Pi}}$ and $F_{\Pi}$

In order to present the branching ratios of different possible channels in
terms of parameters of the model, we are bound to discuss physically sensible
parameters to explore. This model has four parameters that can be rearranged
to three dimensionless parameters for a given $\Lambda$ value:
$(\Lambda,m_{F},F_{\Pi},m_{{}_{\Pi}})\rightarrow(m_{F}/\Lambda,F_{\Pi}/m_{{}_{\Pi}},m_{{}_{\Pi}}/m_{F}).$

In the cross section, decay rate and branching ratios of the composite boson
case, $m_{F}$ is not relevant and we assume $m_{F}=0.8\Lambda$ or
$m_{F}=\Lambda$ as adopted in the previous article [61]. The parameters
reduces to $(F_{\Pi}/\Lambda,m_{{}_{\Pi}})$ or
$(g_{{}_{Y}}^{2},m_{{}_{\Pi}}/F_{\Pi})$. In the plot for the cross section, We
may consider the range $m_{{}_{\Pi}}$ from $0.0$ TeV and $10$ TeV as Figure 5
and $g_{{}_{Y}}^{2}=0.02,0.1,0.5$. In the plot of branching ratios, we
consider to use $(g_{{}_{Y}}^{2},m_{{}_{\Pi}}/F_{\Pi})$, plotting
$m_{{}_{\Pi}}/F_{\Pi}$ from $0.1$ to $1.5$ and $g_{{}_{Y}}^{2}=0.02,0.1,0.5$
???. These cover the parameter space adopted in the previous article [61].
Observe that as indicated in Eq. 127, the
$\Pi^{0}_{u,d}\rightarrow\tilde{G}\tilde{G}^{\prime}$ dominates at larger
values $m_{{}_{\Pi}}/F_{\Pi}$, as also shown in Fig. 5 (centre column) of the
previous article [61].

The ratio $m_{F}/\Lambda<1$ ($m_{\Pi}/\Lambda<1$) of the composite fermion
(boson) mass and the basic composite scale $\Lambda$ gives us an insight into
the dynamics of composite fermion (boson) formation. In addition, as the
parameters $m_{\Pi}$ and $F_{\Pi}$ represent the same dynamics of composite
boson formation we use the $F_{\Pi}/m_{\Pi}$ ratio. Finally, to take into
account the feature that a composite fermion $F$ is composed by a composite
boson $\Pi$ and an elementary SM fermion, we adopt the ratio $m_{\Pi}/m_{F}<1$
as a parameter. As a result, for given $\sqrt{s}$ and $\Lambda$ values, we
have three parameters $m_{F}\approx\Lambda$, $F_{\Pi}/\Lambda$, and
$m_{\Pi}/m_{F}$ to represent the results of the possible branching ratios.

###  X.7 Wide jets and photons from composite bosons $\Pi^{0}_{u,d}$
production and decay

I try to write some topologies here. Apart from $\Pi\rightarrow qq^{\prime}$
that ends with two single jets. we discuss the channels of $\Pi^{0}_{u,d}$
decay into two SM gauge bosons $\tilde{G}\tilde{G}^{\prime}$ in Sec. (IX.1).
We have following three catalogs and we can calculate the branching ratios to
see which channel is relevant and experimentally interesting

The processes with photon in final state are

  1. (i)

Two back-to-back photons (diphoton) for the channel (50)
${\mathcal{B}}=\Gamma_{\Pi\rightarrow\gamma\gamma}/\Gamma_{\rm tot}$;

  2. (ii)

back-to-back photon and a wide jet for the channel (51) are final states.

The processes with $W^{\pm}$ or $Z^{0}$ in final state are, two back-to-back
boosted gauge bosons $Z^{0}$ (52) or $W^{\pm}$ (53) are high-energetic boosted
intermediate states, most probably leading to:

  1. (i)

a back-to-back wide dijet of four quarks for $W^{\pm}$ and $Z^{0}$ hadronic
decays;

  2. (ii)

back-to-back lepton-pair beams of four leptons for $W^{\pm}$ and $Z^{0}$
leptonic decay;

  3. (iii)

a wide jet and a lepton-pair beam in the combination of cases (i) and (ii).
Instead, two back-to-back photons (diphoton) for the channel (50), and back-
to-back photon and a wide jet for the channel (51) are final states.

The processes (LABEL:pgugu) with two gluons $\tilde{g}$ in final state are,

  1. (i)

two gluons (LABEL:pgugu) back-to-back jets, does it fuse to a Higgs then decay
to two photons in finite state ?;

Which channel is more relevant for detections, depending not only on
theoretical branching ratios in principle, but also on experimental
measurements in practice.

The widths of these $\Pi^{0}$-decay channels are
$\Gamma_{\Pi^{0}\rightarrow\tilde{G}\tilde{G}^{\prime}}$ (120-127), and the
total $\Pi^{0}$-decay width is $\Gamma_{\rm tot}$ (128). The branching ratios
are Eqs. (140) and (141). As a result, in $pp$ collision the cross section of
the $\Pi^{0}$ production and decay to two back-to-back standard jets and/or
gauge bosons is (143).

The neutral composite boson $\Pi^{0}$ mass can be inferred by the invariant
mass of its resonance location ${\mathcal{M}}_{\rm
all~{}channels}=m_{\Pi^{0}}$. The invariant mass
${\mathcal{M}}_{\gamma\gamma}$ of the diphoton channel (50) gives the
constrain $m_{\Pi^{0}}\gtrsim{\mathcal{M}}_{\gamma\gamma}$. The invariant mass
${\mathcal{M}}_{\Pi^{\pm,0}\rightarrow jj}$ of the dijet even sets the
constrain on the composite boson mass $m_{{}_{\Pi^{\pm,0}}}$. Due to the
smallness of gauge interactions and quark-family mixing (off-diagonal elements
of the CMK matrix), the branching ratios of $\Pi^{\pm}$ decay to other quark
families are neglected, and $m_{\Pi^{\pm}}\approx m_{\Pi^{0}}$.

##  XI Search for $\Pi$ at the LHC

After having discussed the production and decay of $\Pi$ and its cross-
section, width, and decay branching ratio, we now examine these results in
terms of parameters of the model and derive the possible final states and
their topologies, highlighting their impact to the current program of beyond
SM searches at the LHC.

###  XI.1 Branching ratios and topology of $\Pi$ with respect to model
parameters

In order to present the branching ratios of different possible channels in
terms of parameters of the model, we are bound to discuss physically sensible
parameters to explore. This model has four parameters that can be rearranged
to three dimensionless parameters for a given $\Lambda$ value:
$(\Lambda,m_{F},F_{\Pi},m_{\Pi})\rightarrow(m_{F}/\Lambda,F_{\Pi}/m_{\Pi},m_{\Pi}/m_{F}).$

The ratio $m_{F}/\Lambda<1$ ($m_{\Pi}/\Lambda<1$) of the composite fermion
(boson) mass and the basic composite scale $\Lambda$ gives us an insight into
the dynamics of composite fermion (boson) formation. In addition, as the
parameters $m_{\Pi}$ and $F_{\Pi}$ represent the same dynamics of composite
boson formation we use the $F_{\Pi}/m_{\Pi}$ ratio. Finally, to take into
account the feature that a composite fermion $F$ is composed by a composite
boson and an elementary SM fermion, we adopt the ratio $m_{\Pi}/m_{F}<1$ as a
parameter. As a result, for given $\sqrt{s}$ and $\Lambda$ values, we have
three parameters $m_{F}/\Lambda$, $F_{\Pi}/m_{\Pi}$, and $m_{\Pi}/m_{F}$ to
represent the results of the possible branching ratios. Figure LABEL:AllBr
shows three sets of plots for the branching ratios of
$F\rightarrow\bar{f}qq^{\prime}$ and $F\rightarrow\bar{f}\Pi$ (left column),
$\Pi\rightarrow qq^{\prime}$ and $\Pi\rightarrow G\tilde{G}^{\prime}$ (center
column), and the full decay chains $F\rightarrow\bar{f}qq^{\prime}$,
$F\rightarrow\bar{f}\Pi\rightarrow\bar{f}qq^{\prime}$, and
$F\rightarrow\bar{f}\Pi\rightarrow\bar{f}G\tilde{G}^{\prime}$ (right column).
These branching ratios are plotted with respect to the $m_{\Pi}/m_{F}$ ratio
and for the values of $F_{\Pi}/\,m_{\Pi}=0.8$, $F_{\Pi}/\,m_{\Pi}=5$,
$F_{\Pi}/\,m_{\Pi}=15$. Note that the branching ratios of $F\rightarrow
fqq^{\prime}$ and $F\rightarrow f\Pi$ (left column) do not depend on
$m_{F}/\Lambda$, which can be seen from Eqs. (LABEL:decayE0,LABEL:fep) and
(LABEL:gtfeqq). For the $\Pi^{\pm}$ decay, the channel (119) is unique, so the
branching ratio is one, independent of $m_{F}/\Lambda$. Whereas the
$\Pi^{0}_{u,d}$ decays also to $G^{\prime}G$, see Eqs. (140) and (141), and
the branching ratio depends on $m_{F}/\Lambda$. However, in the regime of
$m_{\Pi}/m_{F}$ we consider, the branching ratio of $\Pi^{0}_{u,d}\rightarrow
G^{\prime}G$ is very small and negligible, compared with the branching ratio
of $\Pi^{0}_{u,d}\rightarrow q^{\prime}q$. Therefore, our results of branching
ratio of $F$ decay presented in Fig. LABEL:AllBr are independent of the
parameter $m_{F}/\Lambda$. As a result, regarding the branching ratio and
topologies of $F$ decay, the model effectively depends only on two parameters
$F_{\Pi}/m_{\Pi}$ and $m_{\Pi}/m_{F}$. However, the cross section of $F$
production depends on the parameter $m_{F}/\Lambda$, as indicated in Eq.
LABEL:csE0.

In Fig. LABEL:AllBr, it is shown that the branching ratios $F\rightarrow
fq\bar{q}~{}{\rm direct}$ (LABEL:Eeqqdirect) and $F\rightarrow f\Pi\rightarrow
fq\bar{q}~{}{\rm indirect}$ (LABEL:Eepqq) tend to swap each other for
different values of $F_{\Pi}/m_{\Pi}$. Increasing the coupling $F_{\Pi}$ (and
the ratio $F_{\Pi}/m_{\Pi}$) of composite boson $\Pi$ to its two constituents
($q\bar{q}$), see contact interactions (16,17,18), the branching ratios
$F\rightarrow f\Pi\rightarrow fq\bar{q}~{}{\rm indirect}$ (LABEL:Eepqq)
becomes dominant over the branching ratios $F\rightarrow fq\bar{q}~{}{\rm
direct}$ (LABEL:Eeqqdirect). We thus consider two reference cases of
$F_{\Pi}/m_{\Pi}$:

  1. (i)

$F_{\Pi}/m_{\Pi}\lesssim 0.8$ where $F\rightarrow fq\bar{q}~{}{\rm direct}$
dominates. We have verified that for the value 0.8 the direct production
dominates by at least a factor 10 over the production with the $\Pi$ for all
values of $m_{\Pi}/m_{F}$.

  2. (ii)

$F_{\Pi}/m_{\Pi}\gtrsim 5$ where $F\rightarrow f\Pi\rightarrow
fq\bar{q}~{}{\rm indirect}$ (LABEL:Eepqq) becomes relevant for all values of
$m_{\Pi}/m_{F}$ and dominates above $m_{\Pi}/m_{F}>$0.2 over $F\rightarrow
fq\bar{q}~{}{\rm direct}$. We notice from Fig. LABEL:AllBr that as
$F_{\Pi}/m_{\Pi}$ increases above 15, the decay $F\rightarrow f\Pi\rightarrow
fq\bar{q}~{}{\rm indirect}$ dominates over all possible decays for all values
of $m_{\Pi}/m_{F}$.

The expected topologies related to the phenomenology of $F$ are summarized in
Table 10, considering the two complementary cases of $F_{\Pi}/m_{\Pi}\leq 0.8$
and $F_{\Pi}/m_{\Pi}\geq 15$. These cases are representative of the topology
of $F$ in all the phase space, including the intermediate region
$0.8<F_{\Pi}/m_{\Pi}<15$, in which the decay $F\rightarrow fq\bar{q}~{}{\rm
direct}$ or $F\rightarrow f\Pi\rightarrow fq\bar{q}~{}{\rm indirect}$
dominates depending on the specific value of $F_{\Pi}/m_{\Pi}$ considered. To
this purpose we provide in Appendix LABEL:Br_Fpimpi_1_15_step1 the total decay
branching ratios corresponding to Fig. LABEL:AllBr (right column) for values
of $F_{\Pi}/m_{\Pi}$ between 1 and 14, increasing in step of 1. We further
outline that the value 0.2 for $m_{\Pi}/m_{F}$ that separates the boosted from
the resolved topology when $F$ decays through a $\Pi$ is indicative and may
vary based on the mass of $F$. We have verified, using CalcHEP, that for a
mass of $F$ of 1 TeV, the two quarks decaying from the $\Pi$ originating from
$F$ are indeed within a $\Delta R(q,\bar{q}^{\prime})$ below 0.8, which
determines the cone size of a large-radius jets suitable for boosted jet at
the LHC experiments. For a mass of $F$ of 7 TeV instead, the value 0.2 is
lowered to 0.15 to guarantee $\Delta R(q,\bar{q}^{\prime})<0.8$, as for higher
masses of $F$ its width increases.

$F_{\Pi}/m_{\Pi}$ | $m_{\Pi}/m_{F}$ | Channel | Resonances | Topology | Experimental features |
---|---|---|---|---|---|---
15 | $[\sim 0.2,\sim 1]$ | $fF\rightarrow f(\bar{f}\Pi)\rightarrow f(\bar{f}(qq^{\prime}))$ | $F,\Pi$ | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | identification of $\Pi$ and $F$ |
| $\leq 0.2$ | $fF\rightarrow f(\bar{f}\Pi)\rightarrow f(\bar{f}(qq^{\prime}))$ |  $F$, $\Pi$ | Boosted | identification of $F$; |
|  |  |  |  |  $\Pi$ large-radius jet: |
|  |  |  |  | 2-prong, no V boson tag |
$\leq$ 0.8 | [0,1] | $fF\rightarrow f(\bar{f}qq^{\prime})$ | $F$ | Fully resolved | same of $F_{\Pi}/m_{\Pi}$ = 10 |

Table 10: Summary of the relevant channels where $F$ can decay considering two
complementary values of $F_{\Pi}/m_{\Pi}$ and the ranges of $m_{\Pi}/m_{F}$
that correspond to different topologies of $F$. Similar considerations on the
channels where $F$ can decay and its topology apply to all values of
$F_{\Pi}/m_{\Pi}$, depending on the value of $F_{\Pi}/m_{\Pi}$, the value of
$m_{\Pi}/m_{F}$, and the branching ratios of $F\rightarrow fq\bar{q}~{}{\rm
direct}$ and $F\rightarrow f\Pi\rightarrow fq\bar{q}~{}{\rm indirect}$.

$f$ | $F$ | Topology | Final state | LHC searches | Features not exploited in LHC searches |
---|---|---|---|---|---|---
$e$ | $E$ | Fully resolved | $e^{\pm}(e^{\mp}qq^{\prime})$ | [28, 49] |  $E$ identification |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $e^{\pm}(e^{\mp}(qq^{\prime}))$ | [53, 49] |  $E,\Pi$ identification |
|  | Boosted | $e^{\pm}e^{\mp}J$ | [28] | 2-prong, no V boson tag, boosted $\Pi$ decay |
$\mu$ | $M$ | Fully resolved | $\mu^{\pm}(\mu^{\mp}qq^{\prime})$ | [28, 49] |  $M$ identification |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $\mu^{\pm}(\mu^{\mp}(qq^{\prime}))$ | [53, 49] |  $M,\Pi$ identification |
|  | Boosted | $\mu^{\pm}\mu^{\mp}J$ | [28] | 2-prong, no V boson tag, boosted $\Pi$ decay |
$\tau$ | $\mathcal{T}$ | Fully resolved | $\tau^{\pm}(\tau^{\mp}qq^{\prime})$ | [50] |  $\mathcal{T}$ identification |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $\tau^{\pm}(\tau^{\mp}(qq^{\prime}))$ | [50] |  $\mathcal{T},\Pi$ identification |
|  | Boosted | $\tau^{\pm}\tau^{\mp}J$ | n/a |  |
$\nu$ | $N$ | Fully resolved | $\nu(\nu qq^{\prime})$ | [51, 52] |  $N$ identification |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $\nu(\nu(qq^{\prime}))$ | [51, 52] |  $N,\Pi$ identification |
|  | Boosted | $\nu\nu J$ | [56] | 2-prong, no V boson tag, boosted $\Pi$ deacy |
$j$ | $J$ | Fully resolved | $j(jqq^{\prime})$ | n/a |  |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $j(j(qq^{\prime}))$ | n/a |  |
|  | Boosted | $jjJ$ | n/a |  |
$c$ | $C$ | Fully resolved | $c(cqq^{\prime})$ | n/a |  |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $c(c(qq^{\prime}))$ | n/a |  |
|  | Boosted | $ccJ$ | n/a |  |
$b$ | $B$ | Fully resolved | $b(bqq^{\prime})$ | n/a |  |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $b(b(qq^{\prime}))$ | n/a |  |
|  | Boosted | $bbJ$ | n/a |  |
$t$ | $T$ | Fully resolved | $t(\bar{t}qq^{\prime})$ | n/a |  |
|  | Resolved w/ $\Pi\rightarrow qq^{\prime}$ | $t(\bar{t}(qq^{\prime}))$ | n/a |  |
|  | Boosted | $t\bar{t}J$ | n/a |  |

Table 11: List of all the signatures foreseen by the model for the different
flavours of $f$, $F$, their topology and final states. In the table “$j$”
indicates any reconstructed jet that originates from a $u,d,s$ quark and “$J$”
a large-radius (cone-size of 0.8) jet that is reconstructed from 2 quarks
produced with low angular separation. The notation w$/\Pi$ indicates the decay
of $F$ through a $\Pi$. The LHC searches for these signatures are reported
along with some comments on the features that are typical of this model and
have not been exploited in the referenced LHC searches. In the table, ”n/a”
indicates the case in which an LHC search interesting the corresponding final
state has not been found.

So far we have presented the studies relying on the first SM generation, i.e.
considering the electron flavour of $F$ ($F=E$). However, these discussions
and calculations are straightforwardly generalized to the second and third SM
generations, and in the text we keep the general notation $F$ instead of $E$.
At the leading order of only contact interactions being taken into account,
the formulae of cross-sections, decay rates and branching ratios of the second
and third generations are the same as those of the first generation, however
composite fermions $F$ and their productions, decay channels and rates are
different, depending on the values of their parameters $m_{F}$, $m_{\Pi}$ and
$F_{\Pi}$, which vary from one SM generation to another. The reasons are that
the effective couplings $\sim g_{*}^{2}/\Lambda^{2}$ and $\sim g_{{}_{Y}}$ of
contact interactions can be different, due to the modifications from the
flavor mixing and gauge interactions. However, we neglect these modifications
in this article, and expect the variations of masses $m_{F}$, $m_{\Pi}$, decay
constant $F_{\Pi}$ to be small. In this sense, there are two basic parameters
$F_{\Pi}/m_{\Pi}$ and $m_{\Pi}/m_{F}$ for each SM generation to be determined
by different channels and their topologies in LHC experiments discussed in
next sections.

###  XI.2 Signatures to search for $F$ at the LHC

We now summarize on the possible signatures with which $F$ can manifest at the
LHC. We remark that $F$ can have the different flavours corresponding to the
Standard Model flavours of $f$ and the fact that these particles are not
necessarily mass degenerate. This implies that they can have different masses
with which they can appear within the energy reach of the LHC, and thus they
have to be searched for independently. Based on the flavours of $f$, we expect
8 different final states to be investigated for the pair $fF$ produced in the
process $pp\rightarrow fF$, which are: $eE,\mu M,\tau\mathcal{T},\nu
N,jJ,cC,bB,tT$. Here, we consider one single channel ($\nu N$) for all the
$\nu$ neutrinos of the Standard Model and one single channel ($jJ$) for the
$u,d,s$ quarks. We notice that the flavour $cC$ is taken separately, because
of the improving performances in c-tagging algorithms at the LHC ([57, 58])
and dedicated searches for new physics with $c$ quarks in the final state
([59, 60]). Moreover, we distinguish the three topologies (resolved with and
without $\Pi$ and boosted) explained in the previous paragraph, so that we
have in total 24 different signatures that have to be considered in order to
pursue a comprehensive search of $F$.

In Table 11 we outline these signatures based on the flavours $f$ and $F$, the
possible topology, the corresponding final states, and the LHC search that, to
the best of our knowledge, could be more sensitive to searching for $F$. In
the last column, we further report on features of $F$ and its decay that have
not been exploited directly in the cited LHC searches and could be used to
improve possible future searches dedicated to $F$. We especially remark that
the $F$ quark flavours appear to be completely unexplored yet and we urge on
the importance of carrying out specific analyses at LHC to investigate it.
This is certainly noteworthy and can have a relevant impact on the beyond the
SM physics program of the LHC.

We notice that, despite possible and with a peculiar signature, the channel
$fF\rightarrow f(\bar{f}\Pi)\rightarrow
f(\bar{f}(\tilde{G}\tilde{G}^{\prime}))$ is negligible since the decay
$\Pi^{0}\rightarrow(\tilde{G}\tilde{G}^{\prime})$ is only relevant for
$F_{\Pi}/\,m_{\Pi}=0.1$ when, however, the decay $F\rightarrow f(\Pi)$ has a
branching ratio close to zero. Because of this we do not include this case in
Table 11 and we point out that this case would become of interest in the case
$F$ is found in one of the possible signatures mentioned above, to study the
nature of the new particle.

We put emphasis on the case of the composite fermions
$F=N^{0},\bar{N}^{0},N^{+},N^{-}$ for the final state $\nu\nu qq^{\prime}$,
where $\nu\bar{\nu}$ stands for the pairs of the SM left-handed neutrino
$\nu^{e}_{L}$ and/or sterile right-handed neutrino $\nu^{e}_{R}$, as the
latter is a candidate of dark-matter particles.

Finally, we acknowledge that the final state $f\bar{f}qq^{\prime}$ is relevant
for a wide range of the parameter space of the model and thus will consider it
in the next section and, in particular, the case of $f=e$ and $F=E$ to derive
limits on the model parameters based on existing LHC results.

##  XII Bounds on the model parameters

Figure 11:  (Color Online). should we plot the cross section $\sigma$ in terms
of $m_{{}_{\Pi}}$ for different values $g^{2}_{Y}=F_{\Pi}/\Lambda$? Recast of
the experimental upper limit on $\sigma(pp\to eeqq^{\prime})$ published in
[28] against the model of composite fermions studied in this article. The
dotted line (solid green line) is the 95% C.L. observed (expected) upper limit
on $\sigma(pp\to eeqq^{\prime})$ as reported in [28]. The solid line (red) is
the theoretical expectation from the model described in this work as given by
Eq. (LABEL:ppcs) for the case $m_{F}/\Lambda=1$, the dashed lines (orange) are
the theoretical expectation from the model for the cases $\Lambda=6,\,9,\,12$
TeV. If $m_{F}/\Lambda=1$ one obtains that the composite fermions of this
model are excluded up to masses $m^{\rm ex}_{F}\approx 4.25$ TeV. Figure 12:
(Color Online). Recast of the experimental upper limit from [28] (dashed line)
and the predicted contour curve at a 5-level statistical significance (solid
line) in the 2-dimensional parameter space $(\Lambda,m_{F})$. The shaded
region denotes unphysical values of the parameters ($\Lambda<m_{F}$). should
we have a plot like this in terms of $m_{{}_{\Pi}}$ and $\Lambda$ or $F_{\Pi}$
?

In this section we provide a discussion of the bounds of the model parameters
taking as a reference the case $F=E$ and the final state $eeqq^{\prime}$ that
has been shown to be sensitive to a wide portion of the parameter space of the
model in the previous section. For this purpose, we recast the 95% confidence
level (C.L.) experimental upper limit on $\sigma(pp\rightarrow eeqq^{\prime})$
using a recent analysis [28] of 2.3 fb-1 data from the 2015 Run II of the LHC
by the CMS collaboration with respect to the predictions of the model of
composite fermions discussed in this article. Note that both electrons and
positrons are collected in the final states of $eeqq^{\prime}$, electrons and
positrons are not distinguished in the data analysis. For the case
$m_{F}/\Lambda=1$ one obtains that the composite fermions of this model are
excluded up to masses $m^{\rm ex}_{F}\approx 4.25$ TeV. This result is shown
in Figure 11, together with the exclusion limits $m^{\rm ex}_{F}\approx
3.3,2.4,1.5$ TeV for $\Lambda$ fixed at 6, 9 and 12 TeV. Figure 12 shows the
exclusion curve, lower (dashed) line, in the 2-dimensional parameter space
$(\Lambda,m_{F})$ for the model obtained via the recasting of the analysis
[28] of 2.3 fb-1 data from the 2015 Run II of the LHC by the CMS
collaboration. Here the regions of the parameter space below the curves are
excluded.

We also performed a study about the potential of a dedicated analysis in the
High Luminosity LHC (HL-LHC) conditions (center of mass energy of 14 TeV and
luminosity of 3 ab-1). We used CalcHEP to generate the processes and DELPHES
[40] to simulate the detector effects. In order to separate the signal from
the background, we selected events with $pt_{e_{1}}\geq$ 180 GeV,
$pt_{e_{2}}\geq$ 80 GeV, $pt_{j_{1}}\geq$ 210 GeV, $m_{ee}\geq$ 300 GeV ($pt$
is the transverse momentum, $e_{1}$ the leading electron, $e_{2}$ the
subleading electron, $j_{1}$ the leading jet and $m_{ee}$ the invariant mass
of the two electrons). Then we evaluated the reconstruction and selection
efficiencies for signal ($\epsilon_{s}$) and background ($\epsilon_{b}$) as
the ratio of the selected and the total generated events. From these
efficiencies, the signal and background cross sections ($\sigma_{s}$,
$\sigma_{b}$) and the integrated luminosity ($L$), it is possible to evaluate
the expected number of events for the signal ($N_{s}$) and the SM background
($N_{b}$) and finally the statistical significance ($S$):

$N_{s}=L\sigma_{s}\epsilon_{s},\quad N_{b}=L\sigma_{b}\epsilon_{b},\quad
S=\frac{N_{s}}{\sqrt{N_{b}}}\,.$ (144)

The $S=5$ contour curve is shown by the upper (solid) line in Figure 12. It
can be used to get indications about the potential for discovery or exclusion
with the experiments at the HL-LHC, showing that there is a wide region of the
model phase space where the existence of the composite fermions can be
investigated; for the case $m_{F}/\Lambda=1$ we can reach masses up to
$\approx 6.2$ TeV. We notice that, despite having considered the case of $F=E$
in this section, it could be inferred that the cross section for $F$, which
should approximately be the same for all its flavours, is sufficient for the
$F$ to appear at the LHC with the statistics already collected at the LHC
experiments, and that is expected by the HL-LHC. Based on this result, we
recommend that the physics program of the LHC consider the new particles
foreseen by this model and their signatures in its investigations.

##  XIII Summary and remarks

Appendix: Leptoquark four-fermion operators

Based on the dynamics of strong four-fermion couplings (1), the composite
bosons $\Pi^{\pm}$ and $\Pi^{0}_{u,d}$ in Table V of Ref. [Leonardi:2018jzn]
are produced in the quark-quark channel,

$\displaystyle G_{\rm
cut}\,\left[(\bar{\psi}^{bi}_{L}u_{Rb})(\bar{u}^{a}_{R}\psi_{Lia})+(\bar{\psi}^{bi}_{L}d_{Rb})(\bar{d}^{a}_{R}\psi_{Lia})\right]+{\rm
h.c.}.$ (145)

Whereas LQ particles produce in the leptoquark channels of three types,

$\displaystyle G_{\rm
cut}\,\Big{[}(\bar{\ell}^{i}_{L}e_{R})(\bar{d}^{a}_{R}\psi_{Lia})+(\bar{\ell}^{i}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}\psi_{Lia})\Big{]}+{\rm
h.c.},$ $\displaystyle G_{\rm
cut}\,\Big{[}(\bar{\ell}^{i}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\ell_{Li})+(\bar{\ell}^{i}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\ell_{Li})\Big{]}+{\rm
h.c.},$ $\displaystyle G_{\rm
cut}\,\Big{[}(\bar{\psi}_{Lia}\nu^{e}_{R})(\bar{\nu}^{e}_{R}\psi_{Lia})+(\bar{\psi}_{Lia}e_{R})(\bar{e}_{R}\psi_{Lia})\Big{]}+{\rm
h.c.},$ (146)

where the elementary fermion fields are the SM doublet
$\ell^{i}_{L}=(\nu^{e}_{L},e_{L})$ and singlet $e_{R}$ with an additional
right-handed neutrino $\nu^{e}_{R}$ for leptons; $\psi_{Lia}=(u_{La},d_{La})$
and $u^{a}_{R},d^{a}_{R}$ for quarks, where the color $a,b$ and
$SU_{L}(2)$-isospin $i$ indexes are summed over. The four-fermion operators
are SM gauge-symmetric and fermion-number conserving. We explicitly give
quark-lepton channels,

$\displaystyle G_{\rm
cut}\,\Big{[}(\bar{e}_{L}e_{R})(\bar{d}^{a}_{R}d_{La})+(\bar{\nu}^{e}_{L}\nu^{e}_{R})(\bar{u}^{a}_{R}u_{La})+(\bar{\nu}^{e}_{L}e_{R})(\bar{d}^{a}_{R}u_{La})\Big{]}+{\rm
h.c.},$ (147) $\displaystyle G_{\rm
cut}\,\left[(\bar{\nu}^{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}\nu^{e}_{L})+(\bar{e}_{L}u^{a}_{R})(\bar{u}^{a}_{R}e_{L})+(\bar{\nu}^{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}\nu^{e}_{L})+(\bar{e}_{L}d^{a}_{R})(\bar{d}^{a}_{R}e_{L})\right]+{\rm
h.c.},$ $\displaystyle G_{\rm
cut}\,\left[(\bar{u}^{a}_{L}\nu^{e}_{R})(\bar{\nu}^{e}_{R}u^{a}_{L})+(\bar{d}^{a}_{L}\nu^{e}_{R})(\bar{\nu}^{e}_{R}d^{a}_{L})+(\bar{u}^{a}_{L}e_{R})(\bar{e}_{R}u^{a}_{L})+(\bar{d}^{a}_{L}e_{R})(\bar{e}_{R}d^{a}_{L})\right]+{\rm
h.c.}.$

Operators (146) or (147) are for the first generation,
$\nu^{e},e,u,d\rightarrow\nu^{\mu},\mu,c,s$ for the second generation, and
$\nu^{e},e,u,d\rightarrow\nu^{\tau},\tau,t,b$ for the third generation.

…….

###### Acknowledgements.

The work of Alfredo Gurrola and Francesco Romeo is supported in part by NSF
Award PHY-1806612. The work of Hao Sun is supported by the National Natural
Science Foundation of China (Grant No.11675033).

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