11institutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016, Uttar Pradesh, INDIA22institutetext: Institute for Particle Physics
Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, U.K

# Effective Operator Bases for Beyond Standard Model Scenarios: An EFT
compendium for discoveries

Upalaparna Banerjee, Joydeep Chakrabortty, Suraj Prakash, Shakeel Ur Rahaman 2
and Michael Spannowsky  upalab, joydeep, surajprk<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is not only conceivable but likely that the spectrum of physics beyond the
Standard Model (SM) is non-degenerate. The lightest non-SM particle may reside
close enough to the electroweak scale that it can be kinematically probed at
high-energy experiments and on account of this, it must be included as an
infrared (IR) degree of freedom (DOF) along with the SM ones. The rest of the
non-SM particles are heavy enough to be directly experimentally inaccessible
and can be integrated out. Now, to capture the effects of the complete theory,
one must take into account the higher dimensional operators constituted of the
SM DOFs and the minimal extension. This construction, BSMEFT, is in the same
spirit as SMEFT but now with extra IR DOFs. Constructing a BSMEFT is in
general the first step after establishing experimental evidence for a new
particle. We have investigated three different scenarios where the SM is
extended by additional (i) uncolored, (ii) colored particles, and (iii)
abelian gauge symmetries. For each such scenario, we have included the most-
anticipated and phenomenologically motivated models to demonstrate the concept
of BSMEFT. In this paper, we have provided the full EFT Lagrangian for each
such model up to mass dimension 6. We have also identified the $CP$, baryon
($B$), and lepton ($L$) number violating effective operators.

††preprint: IPPP/20/37

##  1 Introduction

The Standard Model (SM) of particle physics has been the most successful
theory to describe the dynamics and interactions of sub-atomic particles.
Every prediction that could be made based on the SM Lagrangian has been
substantiated by different experiments, while the converse is not true.
Several observations can not be satisfactorily explained within the Standard
Model framework. Thus, the SM appears to be a theory of fundamental particles
- but not the complete one, i.e. its validity does not extend to arbitrarily
high energy scales. There have been several efforts to extend the SM by
extending its gauge groups and (or) by adding new particles. It is believed
that at very high energies, near the Planck scale, there is a unified gauge
group from where all the low energy physics, including the SM, have descended.
The region between the unified and electroweak scales is potentially populated
with many particles of different mass scales. However, to this day, we are not
confident about the exact nature of the theories beyond the SM (BSM), as we do
not have sufficient experimental data that can help us to isolate a specific
BSM scenario. A plethora of BSM proposals Golling:2016gvc ; Blondel:2019vdq
exist and each of them has its own merits.

For future and ongoing searches of new physics, for example at the LHC, an
important question is whether it is possible to capture the essence of the new
unknown physics using our knowledge about the symmetries and the particle
content. Indeed, this is the underlying idea of an Effective Field theory
(EFT) where the complete Lagrangian is written as:

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{renorm}}+\sum_{i=5}^{n}\sum_{j=1}^{N_{i}}\frac{\mathcal{C}_{j}^{(i)}}{\Lambda^{i-4}}\mathcal{O}_{j}^{(i)}.$
(1)

Here, $(\sum_{j})$ denotes the sum over effective operators ($N_{i}$) each
having mass dimension $i$. $\Lambda$ is the scale of new physics and thus
possesses mass dimension 1. The dimensionless coefficients
$\mathcal{C}_{j}^{(i)}$ are the so-called Wilson coefficients. The second term
in the above equation is the effective Lagrangian ($\mathcal{L}_{EFT}$)
Georgi:1994qn ; Manohar:1996cq ; Kaplan:1995uv ; Kaplan:2005es ;
Burgess:2007pt ; Rothstein:2003mp ; Cohen:2019wxr . The origin of these
effective operators can be understood through two possible mechanisms. First,
if we have prior knowledge about the new physics Lagrangian then we can
suitably integrate out the heavy modes from the UV theory while retaining the
light ones, i.e. infrared (IR) degrees of freedom (DOF). The impact of heavy
DOFs is captured by the effective interactions and their respective WCs.
Second, to capture their effects, we can add the gauge invariant effective
operators in a consistent way. In this case we need to rely only on the on-
shell DOFs and the associated symmetries. It is interesting to note that even
when the exact nature of the UV theory is unknown, this formalism can be very
useful in sensing the integrated out new physics. In this work, we will focus
on this aspect of EFTs Manohar:1996cq ; Burgess:2007pt ; Georgi:1994qn ;
Kaplan:1995uv ; Kaplan:2005es ; Rothstein:2003mp ; Cohen:2019wxr .

Recognising that the SM may only be valid up to a certain high energy scale
beyond which the effects of new physics may become noticeable, the last decade
has seen tremendous progress towards the study of SM physics as an EFT (or
SMEFT) Henning:2014wua ; Jegerlehner:2013cta ; Contino:2016jqw ; Ellis:2014jta
; Berthier:2015gja ; Brivio:2017vri . More precisely, the study of higher
dimensional operators (of mass dimension $\geq$ 5) has attracted a lot of
attention. And these operators have been found to introduce many novel and
interesting predictions. For instance, the only dimension 5 operator shows
lepton number violation and generates a Majorana mass term for the neutrino.
Going to even higher dimensions we even come across predictions of processes
as rare as proton decay Hambye:2017qix ; Fonseca:2018ehk . SMEFT also
encompasses the two paradigms of EFT - the first being the top-down approach
which actually comes about through an interplay of a particular minimal
extension of the SM and through a subset of higher dimension SMEFT operators.
These assume the existence of the minimal extension at some high energy scale
and after integrating out the heavy degree of freedom yields SMEFT effective
operators Henning:2016lyp ; CriadoAlamo:2019kbt ; Huo:2015exa ; Chiang:2015ura
; Ellis:2016enq ; Fuentes-Martin:2016uol . A number of computational tools
such as CoDeX Bakshi:2018ics , Wilson Aebischer:2018bkb , DsixTools
Celis:2017hod , WCxf Aebischer:2017ugx , MatchingTools Criado:2017khh  have
been developed to automatise this procedure. The second one, i.e., the bottom-
up approach is concerned with the construction of complete and independent
operator sets at various mass dimensions based on group-theoretic ideas
Henning:2015alf ; Henning:2017fpj ; Lehman:2015via ; Lehman:2015coa ;
Fonseca:2019yya . Complete and independent sets of SMEFT operators have been
constructed for mass dimensions 6 Grzadkowski:2010es , 7 Lehman:2014jma , 8
Murphy:2020rsh ; Li:2020gnx , and 9 Li:2020xlh ; Liao:2020jmn . Several
ingeniously built modern tools such as GrIP Banerjee:2020bym , BasisGen
Criado:2019ugp , Sym2Int Fonseca:2017lem , ECO Marinissen:2020jmb , and DEFT
Gripaios:2018zrz  have made the construction of higher dimensional operators
straightforward and convenient. The operators obtained by integrating out the
heavy fields from several different SM extensions turn out to be overlapping
subsets of this complete set. Thus, SMEFT provides a common ground to encode
the predictions of different new physics models. At the same time, some of the
higher dimensional operators also provide non-leading order contributions to
the predictions of the SM itself, thus enhancing the precision of theoretical
calculations Banerjee:2013apa ; Hays:2018zze ; Dedes:2018seb .

It is worth noting that the SMEFT construction assumes that new physics
appears at a particular scale and all the non-SM particles are degenerate. A
completely degenerate spectrum in the UV regime of a new theory, however, is
very unlikely. Instead, for a non-degenerate spectrum, there will be a non-SM
degree of freedom with a small mass. If such a BSM particle is light enough to
be kinematically accessible, and couples strongly to the SM DOFs, it may be
counted as an IR DOF along with the SM ones, while the rest of the new
particles would be heavy enough to be integrated out. SMEFT is not designed to
capture such a scenario. As the on-shell IR DOFs are now extended, one has to
compute the new set of effective operators in addition to the SMEFT ones, thus
leading to a new effective operator basis which can be referred to as BSMEFT.
This is the underlying principle behind the Effective Field Theoretic
reformulation of several popular scenarios. Higher mass dimension operators
have been constructed for diverse scenarios such as the extension of SM by a
doubly charged scalar King:2014uha , the Two Higgs Doublet Model
Anisha:2019nzx ; Trautner:2018ipq ; Crivellin:2016ihg ; Karmakar:2017yek ;
Birch-Sykes:2020btk , and the Minimal Left Right Symmetric Model
Anisha:2019nzx . Neutrino mass models are now being studied under the
framework of $\nu$SMEFT and operators of mass dimensions 6 Elgaard-
Clausen:2017xkq ; Chala:2020vqp  and 7 Bhattacharya:2015vja ; Liao:2016qyd
have been constructed for the same. The same ideas have also been applied to
low energy (below electroweak scale) models within the framework of LEFT
Jenkins:2017jig ; Jenkins:2017dyc ; Dekens:2019ept  where operators up to mass
dimension 7 have been constructed Liao:2020zyx . These find great utility in
B-physics Aebischer:2017gaw  and dark matter studies Brod:2017bsw .

To conduct a procedural analysis we must start by investigating possible
minimal extensions of the SM, which are mostly phenomenologically motivated.
To capture the interplay of the SM electroweak sector with the new physics
models, one must address a variety of scenarios starting from SM-singlet real
scalar fields Adhikari:2020vqo ; OConnell:2006rsp ; Barger_2008  to higher
dimensional color singlet multiplets. One must also consider the extensions of
the strong sector using colored scalars and fermions Buchmuller:1986zs . These
minimal extensions have been introduced in an attempt to rationalize very
specific observations. It is worth mentioning that there exist multiple UV
complete theories that may end up leading to the same set of IR DOFs after
suitably and partially integrating out heavy DOFs. So, looking into these
minimal extensions, it is indeed difficult to identify the unique parent
theory. For example, if the SM spectrum is extended by a doubly charged scalar
then its UV root will be difficult to ascertain. It can appear either as an
$SU(2)_{L}$ singlet but non-zero hyper-charged complex scalar field or as a
part of higher dimensional representations of the electroweak gauge group
$SU(2)_{L}\otimes U(1)_{Y}$. In such cases, the natural possibility is that
there exists a hierarchy of masses between the doubly charged scalar and the
other components of the multiplet. It is also possible that the whole
multiplet is lighter than the other non-SM fields. Then that should be counted
as the IR DOF while constructing the effective operators.

BSMEFT can be considered to be the first stride in the step by step process of
unraveling a full BSM model. Collision experiments are expected to detect few
non-SM particles first, rather than unveiling the complete spectrum of an
extension to the SM at once. The first reaction after observing a new
resonance will be to build a BSMEFT theory around this particle - as evidenced
in previous occasions of eventually unconfirmed experimental excesses (see
e.g. Franceschini:2015kwy ; Backovic:2015fnp ; Strumia:2016wys ). Thus, the
BSMEFT models we provide can serve as a compendium for complete operator bases
after a new resonance is observed.

To promulgate the idea of BSMEFT our study must encompass several varieties of
models, which is precisely the purpose of this work. We have organized the
paper as follows. First, we have meticulously described a general procedure to
construct invariant operators in section 2. We have highlighted the various
subtleties associated with it by giving suitable example operators. In this
work, we have carefully selected the BSM scenarios to capture the possible
impact of the effective operators on the electroweak and strong sectors. Thus
we have worked with models where SM is extended by additional color singlet
complex scalars and fermions that transform as different $SU(2)_{L}$
representations and also phenomenologically motivated Lepto-Quark scenarios.
We have further adopted an abelian extension of the gauge sector of the SM,
motivated by a gauge-boson dark matter scenario. In section 3 we have enlisted
the complete and independent sets of operators of mass dimensions 5 and 6 for
all these models. We have arranged the operators on the basis of their
constituents and we have specifically highlighted the operators that violate
baryon and lepton numbers. This will help to analyze and pin down which of the
rare processes are more likely to occur for a given BSM scenario. We have
showcased the flavour structures of each class of operators for each such
model.

##  2 Roadmap of invariant operator construction

In calculating the invariant operators, underlying symmetries play a crucial
role. The quantum fields transform under these symmetries according to their
assigned charges. The goal is to find all invariants under these symmetries,
i.e., singlet configurations containing any number of those quantum fields.
The Lagrangian consists of all such configurations. We classify the symmetries
as follows: (i) space-time and (ii) gauge symmetries. In addition to that we
can have certain kinds of imposed and (or) accidental global symmetries. The
requirement of their violation or conservation driven by phenomenological
needs determines the presence or absence of rare operators. In principle, the
Lagrangian ($\mathcal{L}$) can contain an infinite number of such singlet
terms. But not all of them are phenomenologically important. Thus it is
preferred to write down $\mathcal{L}$ as a polynomial of the invariant
operators and the mass dimension is chosen to be the order of that polynomial.
This allows one to keep the terms up to a mass dimension based on the
experimental precision possibly achieved in the ongoing and (or) future
experiments. In the following subsections we will demonstrate the role of
individual symmetries and the issues related to the dynamical nature of these
fields, e.g., equation of motions and integration by parts.

###  2.1 Tackling space-time symmetry: Lorentz Invariance

The quantum fields under consideration have different spins, which are
determined by their transformation properties under the (3+1)-dimensional
space-time symmetry, dictated here by the Lorentz group $SO(3,1)$. In this
work, our primary focus is on the scalar, vector and spinorial representations
of the Lorentz group. The scalars, spin-0 fields, transform trivially, i.e.
they are singlets under the Lorentz group. While the vectors, i.e. spin-1 and
spinors, i.e. spin-1/2 are non-singlet representations under $SO(3,1)$. We
must recall, here, that our prescription for computing the invariant operators
deals with finite-dimensional unitary representations. The Lorentz group being
non-compact does not have finite-dimensional unitary representations. Hence,
we will realize the representations of $SO(3,1)$ in terms of unitary finite-
dimensional representations of its compact form $SU(2)_{L}\times SU(2)_{R}$,
and we will work within the Weyl basis where the gamma matrices take the
following forms:

$\displaystyle\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}_{\alpha\dot{\beta}}\\\
\overline{\sigma}^{\mu\dot{\alpha}\beta}&0\end{pmatrix},\,\gamma^{5}=\begin{pmatrix}\text{-}\mathbb{I}&&0\\\
0&&\mathbb{I}\end{pmatrix}.$ (2)

Here, $\sigma^{\mu}=(\mathbb{I},\sigma^{i})$,
$\overline{\sigma}^{\mu}=(\mathbb{I},-\sigma^{i})$, with $\sigma^{i}$ being
the Pauli spin-matrices and $\mathbb{I}$ is a 2$\times$2 identity matrix. In
this basis, the non-zero spin fields possess definite chirality. In the case
of fermions, we will work with Weyl spinors $\Psi_{L}$ and $\Psi_{R}$ instead
of the Dirac spinors $\Psi$ and $\overline{\Psi}$ which are defined as
Dreiner:2008tw :

$\displaystyle\Psi=\begin{pmatrix}\chi_{\alpha}\\\
\xi^{\dagger\dot{\alpha}}\end{pmatrix},\;\;\overline{\Psi}=\Psi^{\dagger}\gamma^{0}=\begin{pmatrix}\xi^{\alpha}&\chi^{\dagger}_{\dot{\alpha}}\end{pmatrix}.$
(3)

We can define the two component Weyl spinors $\Psi_{L}$ and $\Psi_{R}$ as four
component ones in the following manner111$\Psi_{L,R}$ are obtained from $\Psi$
using the projection operators $\frac{1\mp\gamma^{5}}{2}$, i.e.,
$\Psi_{L}=\frac{1-\gamma^{5}}{2}\Psi$, and
$\Psi_{R}=\frac{1+\gamma^{5}}{2}\Psi$.:

$\displaystyle\Psi_{L}=\begin{pmatrix}\chi_{\alpha}\\\
0\end{pmatrix},\;\;\overline{\Psi}_{L}=\Psi_{L}^{\dagger}\gamma^{0}=\begin{pmatrix}0&\chi^{\dagger}_{\dot{\alpha}}\end{pmatrix},\;\;\Psi_{R}=\begin{pmatrix}0\\\
\xi^{\dagger\dot{\alpha}}\end{pmatrix},\;\;\overline{\Psi}_{R}=\Psi_{R}^{\dagger}\gamma^{0}=\begin{pmatrix}\xi^{\alpha}&0\end{pmatrix}.$
(4)

Following a similar principle, the field strength tensor $X^{\mu\nu}$ and its
dual
$\tilde{X}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}X^{\rho\sigma}$,
transforming under $SO(3,1)$, must be written in terms of representations of
$SU(2)_{L}\times SU(2)_{R}$, i.e., $X_{L,\mu\nu}$ and $X_{R,\mu\nu}$ as:

$\displaystyle X_{L,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(X_{\mu\nu}-i\tilde{X}_{\mu\nu}\right),\hskip
14.22636pt(X_{L})_{\alpha\beta}=\sigma^{\mu}_{\alpha\dot{\beta}}\,\overline{\sigma}^{\nu\dot{\beta}\kappa}\,\epsilon_{\kappa\beta}\,X_{L,\mu\nu},$
$\displaystyle X_{R,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(X_{\mu\nu}+i\tilde{X}_{\mu\nu}\right),\hskip
14.22636pt(X_{R})^{\dot{\alpha}\dot{\beta}}=\overline{\sigma}^{\mu\dot{\alpha}\kappa}\,\sigma^{\nu}_{\kappa\dot{\kappa}}\,\,\epsilon^{\dot{\kappa}\dot{\beta}}\,X_{R,\mu\nu}.$
(5)

To proceed further, we have identified the quantum fields222In our analysis,
we have put the covariant derivative ($\mathcal{D}$) on an equal footing as
the quantum fields. as the representations of $SU(2)_{L}\times SU(2)_{R}$ and
demarcated them by their respective spin values $(j_{L},j_{R})$ as:

$\displaystyle\Phi$ $\displaystyle\equiv$
$\displaystyle\left(0,0\right),\hskip
2.84544pt\Psi_{L}\equiv\left(\frac{1}{2},0\right),\hskip
2.84544pt\Psi_{R}\equiv\left(0,\frac{1}{2}\right),\hskip
2.84544pt\mathcal{D}\equiv\left(\frac{1}{2},\frac{1}{2}\right),\hskip
2.84544ptX_{L}\equiv\left(1,0\right),\hskip
2.84544ptX_{R}\equiv\left(0,1\right).$ (6)

Here, $\Phi$ refers to a scalar, and $\Psi_{L,R}$, $X_{L,R}$ are defined in
Eqns. (4) and (2.1) respectively.

As mentioned earlier, our primary aim is to construct a set of Lorentz
invariant operators $(\mathcal{O})$ using these fields and that can be
mathematically framed as follows:

$\displaystyle\mathcal{O}$ $\displaystyle\equiv$
$\displaystyle\Phi^{p}\times\Psi_{L}^{q_{1}}\times\Psi_{R}^{q_{2}}\times\mathcal{D}^{r}\times
X_{L}^{s_{1}}\times X_{R}^{s_{2}},$ (7) $\displaystyle\implies(0,0)$
$\displaystyle\equiv$
$\displaystyle(0,0)^{p}\times\left(\frac{1}{2},0\right)^{q_{1}}\times\left(0,\frac{1}{2}\right)^{q_{2}}\times\left(\frac{1}{2},\frac{1}{2}\right)^{r}\times\left(1,0\right)^{s_{1}}\times\left(0,1\right)^{s_{2}}.$
(8)

Here $p,q_{1},q_{2},r,s_{1},s_{2}$ are the number of times the different
fields appear in the operator. All these are non-negative integers. The
equivalent relation in terms of mass dimension can be written as:

$\displaystyle[M]^{d}$ $\displaystyle\equiv$
$\displaystyle[M]^{p}\times[M]^{3q_{1}/2}\times[M]^{3q_{2}/2}\times[M]^{r}\times[M]^{2s_{1}}\times[M]^{2s_{2}},$
(9)

and equating mass dimensions on both sides we find

$\displaystyle d$ $\displaystyle=$ $\displaystyle
p+\frac{3}{2}(q_{1}+q_{2})+r+2(s_{1}+s_{2}).$ (10)

Here, $d$ is the mass dimension of the Lorentz invariant operator and that for
fermionic and bosonic fields, and field strength tensors are $3/2$, $1$, and
$2$ respectively333The covariant derivative $\mathcal{D}$ has mass dimension
1.. Similarly, the relation derived from Eqn. (8) can be expressed in terms of
the spin ($j$) as:

$\displaystyle 0$ $\displaystyle\equiv$
$\displaystyle[0]^{p}\oplus[1/2]^{q_{1}}\oplus[0]^{q_{2}}\oplus[1/2]^{r}\oplus[1]^{s_{1}}\oplus[0]^{s_{2}},$
$\displaystyle 0$ $\displaystyle\equiv$
$\displaystyle[0]^{p}\oplus[0]^{q_{1}}\oplus[1/2]^{q_{2}}\oplus[1/2]^{r}\oplus[0]^{s_{1}}\oplus[1]^{s_{2}},$
(11)

or equivalently in terms of $SU(2)$ representations ($2j+1$) as:

$\displaystyle 1$ $\displaystyle\equiv$
$\displaystyle[1]^{p}\otimes[2]^{q_{1}}\otimes[0]^{q_{2}}\otimes[2]^{r}\otimes[3]^{s_{1}}\otimes[1]^{s_{2}},$
$\displaystyle 1$ $\displaystyle\equiv$
$\displaystyle[1]^{p}\otimes[1]^{q_{1}}\otimes[2]^{q_{2}}\otimes[2]^{r}\otimes[1]^{s_{1}}\otimes[3]^{s_{2}}.$
(12)

Here, $[1/2]^{q}$ in Eqn. (2.1) and $[2]^{q}$ in Eqn. (2.1) imply
$\underbrace{\vec{\frac{1}{2}}+\cdots+\vec{\frac{1}{2}}}_{q}$ and
$\underbrace{2\otimes\cdots\otimes 2}_{q}$ respectively. Now simultaneously
solving Eqns. (10), (2.1), and (2.1) we can find Lorentz invariant
operators444These are also the operator classes at a given mass dimension..
The number of possible operator classes keeps on increasing as the mass
dimension increases. In Table 1 we have listed all possible operator classes
up to dimension 6 consisting of $\Phi$, $\Psi_{L}$, $\Psi_{R}$, $X_{L}$,
$X_{R}$, and $\mathcal{D}$. But they are not written in covariant forms which
are necessary for further analysis. Below, we have explicitly shown how the
Lorentz indices must be assigned to the constituent fields to write down the
invariant operator in a covariant form.

dim - $(d)$ | $p$ | $q_{1}$ | $q_{2}$ | $r$ | $s_{1}$ | $s_{2}$ | Class |  dim - $(d)$ | $p$ | $q_{1}$ | $q_{2}$ | $r$ | $s_{1}$ | $s_{2}$ | Class
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 0 | 0 | 0 | 0 | 0 | $\Phi$ | 3 | 3 | 0 | 0 | 0 | 0 | 0 | $\Phi^{3}$
2 | 2 | 0 | 0 | 0 | 0 | 0 | $\Phi^{2}$ | 0 | 2 | 0 | 0 | 0 | 0 | $\Psi_{L}^{2}$
0 | 0 | 0 | 2 | 0 | 0 | $\mathcal{D}^{2}$ | 0 | 0 | 2 | 0 | 0 | 0 | $\Psi_{R}^{2}$
|  |  |  |  |  |  | 1 | 0 | 0 | 2 | 0 | 0 | $\Phi\,\mathcal{D}^{2}$
4 | 4 | 0 | 0 | 0 | 0 | 0 | $\Phi^{4}$ | 4 | 0 | 0 | 0 | 4 | 0 | 0 | $\mathcal{D}^{4}$
1 | 2 | 0 | 0 | 0 | 0 | $\Psi_{L}^{2}\,\Phi$ | 1 | 0 | 2 | 0 | 0 | 0 | $\Psi_{R}^{2}\,\Phi$
0 | 1 | 1 | 1 | 0 | 0 | $\Psi_{L}\,\Psi_{R}\,\mathcal{D}$ | 2 | 0 | 0 | 2 | 0 | 0 | $\Phi^{2}\,\mathcal{D}^{2}$
0 | 0 | 0 | 0 | 2 | 0 | $X_{L}^{2}$ | 0 | 0 | 0 | 0 | 0 | 2 | $X_{R}^{2}$
0 | 0 | 0 | 2 | 1 | 0 | $\mathcal{D}^{2}\,X_{L}$ | 0 | 0 | 0 | 2 | 0 | 1 | $\mathcal{D}^{2}\,X_{R}$
5 | 2 | 2 | 0 | 0 | 0 | 0 | $\Psi_{L}^{2}\,\Phi^{2}$ | 5 | 2 | 0 | 2 | 0 | 0 | 0 | $\Psi_{R}^{2}\,\Phi^{2}$
2-810-16 | 5 | 0 | 0 | 0 | 0 | 0 | $\Phi^{5}$ | 1 | 1 | 1 | 1 | 0 | 0 | $\Psi_{L}\,\Psi_{R}\,\Phi\,\mathcal{D}$
| 0 | 2 | 0 | 2 | 0 | 0 | $\Psi_{L}^{2}\,\mathcal{D}^{2}$ | 0 | 0 | 2 | 2 | 0 | 0 | $\Psi_{R}^{2}\,\mathcal{D}^{2}$
| 1 | 0 | 0 | 0 | 2 | 0 | $\Phi\,X_{L}^{2}$ | 1 | 0 | 0 | 0 | 0 | 2 | $\Phi\,X_{R}^{2}$
| 0 | 2 | 0 | 0 | 1 | 0 | $\Psi_{L}^{2}\,X_{L}$ | 0 | 0 | 2 | 0 | 0 | 1 | $\Psi_{R}^{2}\,X_{R}$
| 1 | 0 | 0 | 4 | 0 | 0 | $\Phi\,\mathcal{D}^{4}$ | 3 | 0 | 0 | 2 | 0 | 0 | $\Phi^{3}\mathcal{D}^{2}$
| 1 | 0 | 0 | 2 | 1 | 0 | $\Phi\,X_{L}\,\mathcal{D}^{2}$ | 1 | 0 | 0 | 2 | 0 | 1 | $\Phi\,X_{R}\,\mathcal{D}^{2}$
6 | 6 | 0 | 0 | 0 | 0 | 0 | $\Phi^{6}$ | 6 | 4 | 0 | 0 | 2 | 0 | 0 | $\Phi^{4}\,\mathcal{D}^{2}$
2 | 0 | 0 | 0 | 2 | 0 | $\Phi^{2}\,X_{L}^{2}$ | 2 | 0 | 0 | 0 | 0 | 2 | $\Phi^{2}\,X_{R}^{2}$
1 | 2 | 0 | 0 | 1 | 0 | $\Psi_{L}^{2}\,\Phi X_{L}$ | 1 | 0 | 2 | 0 | 0 | 1 | $\Psi_{R}^{2}\,\Phi X_{R}$
0 | 0 | 0 | 0 | 3 | 0 | $X_{L}^{3}$ | 0 | 0 | 0 | 0 | 0 | 3 | $X_{R}^{3}$
3 | 2 | 0 | 0 | 0 | 0 | $\Psi_{L}^{2}\,\Phi^{3}$ | 3 | 0 | 2 | 0 | 0 | 0 | $\Psi_{R}^{2}\,\Phi^{3}$
0 | 4 | 0 | 0 | 0 | 0 | $\Psi_{L}^{4}$ | 0 | 0 | 4 | 0 | 0 | 0 | $\Psi_{R}^{4}$
0 | 2 | 2 | 0 | 0 | 0 | $\Psi_{L}^{2}\,\Psi_{R}^{2}$ | 2 | 1 | 1 | 1 | 0 | 0 | $\Psi_{L}\,\Psi_{R}\,\Phi^{2}\,\mathcal{D}$
2-810-16 | 0 | 0 | 0 | 2 | 2 | 0 | $\mathcal{D}^{2}\,X_{L}^{2}$ | 0 | 0 | 0 | 2 | 0 | 2 | $\mathcal{D}^{2}\,X_{R}^{2}$
| 2 | 0 | 0 | 2 | 1 | 0 | $\Phi^{2}\,X_{L}\,\mathcal{D}^{2}$ | 2 | 0 | 0 | 2 | 0 | 1 | $\Phi^{2}\,X_{R}\,\mathcal{D}^{2}$
| 1 | 2 | 0 | 2 | 0 | 0 | $\Psi_{L}^{2}\,\Phi\,\mathcal{D}^{2}$ | 1 | 0 | 2 | 2 | 0 | 0 | $\Psi_{R}^{2}\,\Phi\,\mathcal{D}^{2}$
| 2 | 0 | 0 | 4 | 0 | 0 | $\Phi^{2}\,\mathcal{D}^{4}$ | 0 | 0 | 0 | 2 | 1 | 1 | $\mathcal{D}^{2}\,X_{L}\,X_{R}$
| 0 | 1 | 1 | 1 | 1 | 0 | $\Psi_{L}\,\Psi_{R}\,X_{L}\,\mathcal{D}$ |  | 0 | 1 | 1 | 1 | 0 | 1 | $\Psi_{L}\,\Psi_{R}\,X_{R}\,\mathcal{D}$
Table 1: Lorentz Invariant Operator classes (in Weyl representation) upto mass
dimension 6. For the case of dimensions 5 and 6, only the operator classes
above the dashed line appear in SMEFT. The terms in red are total derivative
terms and are therefore excluded from the Lagrangian. These operator classes
are not all independent.

  * •

Total derivative terms: The Lorentz invariant total derivative operators that
appear up to dimension 6 are of the following forms:

$\displaystyle\mathcal{D}^{2}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{D}_{\mu}\mathcal{D}^{\mu},\,\,\,\Phi\mathcal{D}^{2}\rightarrow\mathcal{D}_{\mu}(\mathcal{D}^{\mu}\Phi),\,\,\,\mathcal{D}^{4}\rightarrow(\mathcal{D}_{\mu}\mathcal{D}^{\mu})^{2},$
$\displaystyle(\mathcal{D}^{2}X_{L}+\mathcal{D}^{2}X_{R})$
$\displaystyle\rightarrow$
$\displaystyle(\mathcal{D}_{\mu}\mathcal{D}_{\nu}X^{\mu\nu}+\mathcal{D}_{\mu}\mathcal{D}_{\nu}\tilde{X}^{\mu\nu}),$
$\displaystyle\Phi\mathcal{D}^{4}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{D}_{\mu}\mathcal{D}_{\nu}\mathcal{D}^{\nu}(\mathcal{D}^{\mu}\Phi),\,\,\,\,\Phi^{2}\mathcal{D}^{4}\rightarrow\mathcal{D}_{\mu}\mathcal{D}^{\mu}(\mathcal{D}^{\nu}\Phi)(\mathcal{D}_{\nu}\Phi).$
(13)

But being total derivatives, they do not leave any impact. Thus they are
suitably removed from the Lagrangian density.

  * •

Operators containing only scalar fields: Spin-0 field ($\Phi$) is a Lorentz
scalar. Therefore the operators

$\displaystyle\Phi,\,\,\,\Phi^{2},\,\,\,\Phi^{3},\,\,\,\Phi^{4},\,\,\,\Phi^{5},\,\,\,\Phi^{6},\cdots$
(14)

which consist of $\Phi$ only are Lorentz invariant.

  * •

Operators containing fermion bi-linears: In the Weyl basis, there exist three
different fermion bi-linears: $\Psi_{L}^{2}$, $\Psi_{R}^{2}$,
$\Psi_{L}\Psi_{R}$. The first two terms can form Lorentz invariant operators
of mass dimension three. The last one appears only as a constituent of higher
dimensional operators, since it transforms as the
$\left(\frac{1}{2},\frac{1}{2}\right)$ representation of $SU(2)_{L}\times
SU(2)_{R}$. These fermion bi-linears can be written in multiple covariant
forms:

$\displaystyle\Psi_{L,R}^{2}$ $\displaystyle\rightarrow$
$\displaystyle\underline{\Psi_{L,R}^{T}\,C\,\Psi_{L,R}},\,\,\underline{\overline{\Psi}_{R,L}\,\Psi_{L,R}},\,\,\Psi^{T}_{L,R}\,C\,\sigma^{\mu\nu}\,\Psi_{L,R},\,\,\overline{\Psi}_{R,L}\,\sigma^{\mu\nu}\,\Psi_{L,R},$
$\displaystyle\Psi_{L}\Psi_{R}$ $\displaystyle\rightarrow$
$\displaystyle\overline{\Psi}_{L}\,\gamma^{\mu}\,\Psi_{L},\,\,\overline{\Psi}_{R}\,\gamma^{\mu}\,\Psi_{R}.$
(15)

Here, $\sigma^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$ and $C$ is the
charge conjugation operator. In the above equation only underlined terms are
Lorentz invariant. The remaining structures combine with other Lorentz non-
singlet terms to form higher dimensional operators. Some of those invariant
structures have been listed below:

$\displaystyle\overline{\Psi}_{L}\,\gamma^{\mu}\,\Psi_{L}\,\,\times\,\,\mathcal{D}_{\mu}\,$
$\displaystyle\equiv$ $\displaystyle\,\Psi_{L}\Psi_{R}\,\mathcal{D},\hskip
14.22636pt\overline{\Psi}_{L}\,\gamma^{\mu}\,\Psi_{L}\,\,\times\,\,\Phi\,\mathcal{D}_{\mu}\,\Phi\,\equiv\,\Psi_{L}\Psi_{R}\,\Phi^{2}\,\mathcal{D},$
$\displaystyle\overline{\Psi}_{L}\,\gamma^{\mu}\,\Psi_{L}\,\,\times\,\,\overline{\Psi}_{R}\,\gamma^{\mu}\,\Psi_{R}\,$
$\displaystyle\equiv$ $\displaystyle\,\Psi_{L}^{2}\Psi_{R}^{2},\hskip
31.2982pt\overline{\Psi}_{R}\,\sigma^{\mu\nu}\,\Psi_{L}\,\,\times\,\,X_{\mu\nu}\,\equiv\,\Psi_{L}^{2}X_{L},$
$\displaystyle\overline{\Psi}_{R}\,\Psi_{L}\,\,\times\,\,\Phi\,$
$\displaystyle\equiv$ $\displaystyle\,\Psi_{L}^{2}\Phi,\hskip
45.52458pt\overline{\Psi}_{R}\,\Psi_{L}\,\,\times\,\,\overline{\Psi}_{R}\,\Psi_{L}\,\equiv\,\Psi_{L}^{4}.$
(16)

  * •

Operators containing Field strength tensors: As $X_{L},X_{R}$ transform as
$(1,0),\,(0,1)$ respectively under $SU(2)_{L}\times SU(2)_{R}$, they form the
following Lorentz scalars: $X_{L}^{2}$, $X_{R}^{2},\,X_{L}^{3},\,X_{R}^{3}$ up
to dimension 6. They can be expressed in terms of
$X_{\mu\nu},\,\tilde{X}_{\mu\nu}\in SO(3,1)$ as:

$\displaystyle X_{L}^{2}\,+\,X_{R}^{2}\,\,$ $\displaystyle\rightarrow$
$\displaystyle\,\,X_{\mu\nu}X^{\mu\nu}+\tilde{X}_{\mu\nu}X^{\mu\nu},$
$\displaystyle X_{L}^{3}\,+\,X_{R}^{3}\,\,$ $\displaystyle\rightarrow$
$\displaystyle\,\,\underline{X^{\mu}{}_{\nu}X^{\nu}{}_{\kappa}X^{\kappa}{}_{\mu}}+\underline{\tilde{X}^{\mu}{}_{\nu}X^{\nu}{}_{\kappa}X^{\kappa}{}_{\mu}}.$
(17)

The tri-linear terms being overall traces of the combination of three
antisymmetric tensors vanish. This method can be adopted to construct higher
dimensional operators, e.g., at dimension 8 we will have:

$\displaystyle
X_{L}^{4}+X_{L}^{2}X_{R}^{2}+X_{R}^{4}\,\rightarrow\,(X_{\mu\nu}X^{\mu\nu})\,(X_{\kappa\lambda}X^{\kappa\lambda})+(\tilde{X}_{\mu\nu}X^{\mu\nu})\,(X_{\kappa\lambda}X^{\kappa\lambda})+(\tilde{X}_{\mu\nu}X^{\mu\nu})\,(\tilde{X}_{\kappa\lambda}X^{\kappa\lambda}).$

The field strength tensor $X_{L/R}$ may combine with other Lorentz non-singlet
objects to form an invariant operator class, e.g.,

$\displaystyle\mathcal{D}^{2}X^{2}_{L,R}$ $\displaystyle\equiv$
$\displaystyle(\mathcal{D}_{\mu}X^{\mu\nu})^{2},\,\,\,\,\,X_{L,R}\Phi^{2}\mathcal{D}^{2}\equiv(\mathcal{D}_{\mu}X^{\mu\nu})(\Phi\mathcal{D}_{\nu}\Phi),\,\,\,\,\,\mathcal{D}^{2}X_{L}X_{R}\equiv(\mathcal{D}_{\nu}\mathcal{D}_{\mu}X^{\mu\kappa}\tilde{X}_{\kappa}^{\nu}).$

###  2.2 Role of Gauge Symmetry

So far we have discussed the possible structures of the operators which are
constituted of quantum fields with spin $0,\,1/2,\,1$ only and taking only the
space-time symmetry into account. In a realistic particle physics model, there
are additional local and (or) global internal symmetries. As a result of this,
there could be particles of different internal quantum numbers but possessing
the same spin. Such fields will be equivalent to each other with respect to
the Lorentz symmetry. But based on their internal charges, these fields will
combine in a variety of ways leading to different sub-categories of operators
within the same class. Thus, while Lorentz invariance provides us a list of
possible operator classes, it is the internal symmetry which ultimately
decides which combinations are permitted and which ones are not. This can be
elucidated through the most popular example: the Standard Model gauge
symmetry. Looking into its particle content and their quantum numbers in Table
29, it is evident that many of the operator classes in Table 1 do not respect
the SM gauge symmetry. Thus they are excluded from the operator basis. Here,
we have systematically explained the impact of internal gauge symmetries.

  * •

$\Phi^{n}$ operator class with integer $n$: The SM Higgs transforms under
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ as $(1,2,1/2)$. Thus, the
operators containing an odd number of $H$ fields violate both $SU(2)$ and
$U(1)$ symmetries. If $n$ is an even integer then all the operators of the
forms $H^{n}$, $(H^{\dagger})^{n}$, and
$H^{\frac{n}{2}}(H^{\dagger})^{\frac{n}{2}}$ are $SU(2)$ invariant. But only
the $(H^{\dagger}H)$ and its powers are SM singlets. In BSM scenarios that
contain multiple scalars, we may end up with more intricate structures. For
example, if we add an $SU(2)$ triplet scalar $\Delta$ with hypercharge of +1,
there will be an invariant operator
$H^{T}\,\Delta^{\dagger}\,H\in\Phi^{3}$-class.

  * •

Operators involving field strength tensors: Lorentz invariance allows us to
construct terms containing an even number of field strength tensors. But the
internal symmetry prevents their mixing, e.g., in SM there are no cross-terms
between $B_{\mu\nu}$, $W_{\mu\nu}^{I}$ and $G_{\mu\nu}^{A}$. But this need not
be true for certain BSM scenarios. For example, if there are multiple abelian
symmetries, then we can expect some mixing in the gauge kinetic sector.
Looking into the Lorentz symmetry only, the term involving tri-linear field
strengths vanishes due to its anti-symmetric structure. But internal non-
abelian gauge symmetries allow such terms at the dimension 6 level. Within the
SM, $B^{\mu}{}_{\nu}B^{\nu}{}_{\kappa}B^{\kappa}{}_{\mu}$ is absent but
$f^{ABC}\,G^{A\mu}_{\nu}\,G^{B\nu}_{\kappa}\,G^{C\kappa}_{\mu}$ and
$\epsilon^{IJK}\,W^{I\mu}_{\nu}\,W^{J\nu}_{\kappa}\,W^{K\kappa}_{\mu}$ possess
non-vanishing contributions. Here, the anti-symmetric tensors $f^{ABC}$ and
$\epsilon^{IJK}$ are $SU(3)$ and $SU(2)$ structure constants respectively.

  * •

Operators containing of bi-linear fermion fields: Lorentz invariance allows
fermion mass terms of the forms $\Psi_{L}^{2}$ (Majorana) and
$\overline{\Psi}_{L}\,\Psi_{R}$ (Dirac). But in the SM, left and right chiral
fermions are on a different footing. Hence, these terms are forbidden by the
internal symmetries. Further, the quantum numbers of the fields allow the
couplings of fermion bi-linears with the Higgs scalar in the form of the
Yukawa interactions - $\overline{L}\,e\,H$, $\overline{Q}\,d\,H$ and
$\overline{Q}\,u\,i\tau_{2}\,H^{*}$. In addition, the $SU(3)$ symmetry
prevents the appearance of terms like $\overline{L}\,u$, $\overline{L}\,d$ and
$\overline{Q}\,e$ 555These terms appear as constituents of certain dimension 9
operators.. Also, the operator class
$(\overline{\Psi}_{L}\,\sigma_{\mu\nu}\,\Psi_{R})\,\Phi\,X^{\mu\nu}$ appears
at mass dimension 6. The choice of $X^{\mu\nu}$ and $\Psi$’s is fixed by the
internal symmetries. There are fermion bi-linears which are not Lorentz scalar
but may appear in higher mass dimensional operator class
$(\overline{\Psi}\,\gamma_{\mu}\,\Psi)\,(\overline{\Psi^{\prime}}\,\gamma^{\mu}\,\Psi^{\prime})$
666The fermion fields $\Psi$ and $\Psi^{\prime}$ need not be same always..

###  2.3 Removal of redundancies and forming Operator basis

So far we have learnt how to compute the invariant operators of any mass
dimension based on the space-time and internal symmetries. But we must keep in
mind the fact that these operators need to satisfy another criteria to be
phenomenologically relevant. The operators at each mass dimension must form a
basis, i.e., they must be mutually independent. Thus it is necessary to remove
all the redundancies, if any, to compute the operator basis. In this
construction, we have noted three different ways in which the operators can be
interrelated: (i) integration by parts (IBP), (ii) equation of motion (EOM),
and (iii) identities of symmetry generators. Here, we have discussed these
sources of redundancies briefly with examples based on SMEFT and beyond.

#### Integration by Parts (IBP)

In our prescription, the covariant derivative ($\mathcal{D}_{\mu}$)
participates in the operator construction in a similar way as the quantum
fields. Due to the distributive property of $\mathcal{D}_{\mu}$ and
incorporating integration by parts (IBPs), two or more invariant operators may
be related to each other by a total derivative. As we know such a term in the
Lagrangian has no role to play, thus it can be removed. Therefore the multiple
operators can not be treated independently and only one of them should be
included in the operator basis. This duplication due to IBP occurs among
different operators belonging to the same operator class. For example, at mass
dimension 6, the operator $\Psi_{L}\Psi_{R}\,\Phi^{2}\mathcal{D}$ can be
recast in the following form:

$\displaystyle
i\mathcal{D}_{\mu}\,(\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,\Phi^{\dagger}\Phi)$
$\displaystyle=$
$\displaystyle\overline{\Psi}_{L,R}\,\gamma^{\mu}i\mathcal{D}_{\mu}\,\Psi_{L,R}\,\Phi^{\dagger}\Phi-\overline{\Psi}_{L,R}\,\gamma^{\mu}i\overleftarrow{\mathcal{D}_{\mu}}\,\Psi_{L,R}\,\Phi^{\dagger}\Phi$
(18)
$\displaystyle+\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,\Phi^{\dagger}i\mathcal{D}_{\mu}\Phi-\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,\Phi^{\dagger}i\overleftarrow{\mathcal{D}_{\mu}}\Phi$
$\displaystyle=$
$\displaystyle(\overline{\Psi}_{L,R}\,\gamma^{\mu}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Psi_{L,R})\,\Phi^{\dagger}\Phi+\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Phi).$

Here, $i\overleftrightarrow{\mathcal{D}}_{\mu}\equiv
i\mathcal{D}_{\mu}-i\overleftarrow{\mathcal{D}}_{\mu}$ has been introduced to
combine the first two and the last two operators to form
$(\overline{\Psi}_{L,R}\,\gamma^{\mu}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Psi_{L,R})\,\Phi^{\dagger}\Phi$
and
$\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Phi)$
which are self-hermitian. It is evident from Eqn. (18), that these operators
are related to each other by a total derivative term
$\mathcal{D}_{\mu}\,(\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,\Phi^{\dagger}\Phi)$.
So, in the operator basis we will include only one of them. Here, our choice
of the independent operator will be the one where the derivative acts on the
scalar field. This is because the latter structure where the derivative acts
on the fermions is related to other operators through equations of motion. We
will justify this choice in the following section.

#### Equation of Motion (EOM)

The quantum fields representing the particles are dynamical in nature and each
of them satisfies their respective equation of motion. It has been noted that
two or more operators may be related to each other through the EOMs of the
involved fields along with the IBPs Henning:2017fpj ; Grzadkowski:2010es .
Unlike the previous case, the EOMs can relate operators belonging to different
classes. We have explained how EOM leads to redundancy using a few examples:

  * •

$\boxed{\Psi_{L}\Psi_{R}\,\Phi^{2}\mathcal{D}}$ : In the Weyl basis we can
have two possible covariant structures for this operator:
$(\overline{\Psi}_{L,R}\,\gamma^{\mu}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Psi_{L,R})\,\Phi^{\dagger}\Phi$
and
$\overline{\Psi}_{L,R}\,\gamma^{\mu}\,\Psi_{L,R}\,(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Phi)$.
We have already noted that these two operators differ from each other by a
total derivative. There we have further mentioned that we have selected the
operator where the derivative is acting on the scalars. The reason behind that
choice is that after incorporating the EOMs of $\Psi$ or its conjugate
$\overline{\Psi}$, this operator reduces to an operator belonging to
$\Psi_{L,R}^{2}\Phi^{3}$ class:

$\displaystyle(\overline{\Psi}_{L,R}\,\gamma^{\mu}i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Psi_{L,R})\,\Phi^{\dagger}\Phi\,\,\propto\,\,\overline{\Psi}_{L,R}\,\Psi_{R,L}\,\Phi\,(\Phi^{\dagger}\Phi)\,\,\equiv\Psi_{L,R}^{2}\,\Phi^{3}.$
(19)

  * •

$\boxed{\Psi_{L,R}^{2}\,\Phi\,D^{2}}$ : The unique covariant form of this
operator is $(\overline{\Psi}_{L}\,\Psi_{R})\,\mathcal{D}^{2}\,\Phi$. After
implementing the EOM of the scalar field:
$\mathcal{D}^{2}\,\Phi=c_{1}\,\Phi+c_{2}\,\Phi\,(\Phi^{\dagger}\Phi)+c_{3}\,\overline{\Psi}_{R}\,\Psi_{L}$,
this operator can be reduced in the following form:

$\displaystyle(\overline{\Psi}_{L}\Psi_{R})\,\mathcal{D}^{2}\,\Phi$
$\displaystyle=$ $\displaystyle
c_{1}\,\underbrace{(\overline{\Psi}_{L}\Psi_{R}\,\Phi)}_{\text{dim-4
term}}+c_{2}\,(\overline{\Psi}_{L}\Psi_{R}\,\Phi)\,(\Phi^{\dagger}\Phi)+c_{3}\,(\overline{\Psi}_{L}\Psi_{R})\,(\overline{\Psi}_{R}\Psi_{L}),$
(20)

with $c_{1},c_{2}$ and $c_{3}$ being complex numbers. Thus, the operator class
$\Psi^{2}_{L,R}\,\Phi\,\mathcal{D}^{2}$ can be expressed as a linear
combination of two other dimension 6 classes $\Psi_{L,R}^{2}\Phi^{3}$ and
$\Psi_{L}^{2}\Psi_{R}^{2}$, and therefore is excluded from the set of
independent operators.

  * •

$\boxed{\mathcal{D}^{2}\,X^{2}_{L,R},\,\mathcal{D}^{2}\,X_{L}\,X_{R}}$ : The
possible covariant form of the operators are (i)
$(\mathcal{D}_{\mu}X^{\mu\nu})^{2}$, (ii)
$(\mathcal{D}_{\mu}X^{\mu\nu})(\mathcal{D}_{\mu}\tilde{X}^{\mu\nu})$, and
(iii) $(\mathcal{D}_{\mu}\tilde{X}^{\mu\nu})^{2}$. It is interesting to note
that after implementing the EOM of field strength tensors:

$\displaystyle\mathcal{D}_{\mu}\tilde{X}^{\mu\nu}=0,\hskip
19.91684pt\mathcal{D}_{\mu}X^{\mu\nu}=\overline{\Psi}_{L,R}\gamma^{\nu}\,\Psi_{L,R}+\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\nu}\Phi,$
(21)

the last two structures (ii) and (iii) identically vanish. The very first
operator can be rewritten either as:

$\displaystyle(\mathcal{D}_{\mu}X^{\mu\nu})^{2}$ $\displaystyle=$
$\displaystyle
a_{1}(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})(\mathcal{D}_{\mu}\,X^{\mu\nu})+a_{2}(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\nu}\Phi)(\mathcal{D}_{\mu}\,X^{\mu\nu}),$
(22)

or as:

$\displaystyle(\mathcal{D}_{\mu}X^{\mu\nu})^{2}$ $\displaystyle=$
$\displaystyle
b_{1}\,(\overline{\Psi}_{L,R}\,\gamma^{\nu}\,\Psi_{L,R})^{2}+b_{2}\,(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\nu}\Phi)^{2}+b_{3}(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\nu}\Phi)(\overline{\Psi}_{L,R}\,\gamma^{\nu}\,\Psi_{L,R}).$
(23)

Here, $a_{i},b_{i}$ are complex numbers. Thus we can generate operators
belonging to
$\Phi^{4}\mathcal{D}^{2},\,\Psi^{4},\,\Phi^{2}\Psi^{2}\mathcal{D}$ starting
from $\mathcal{D}^{2}X^{2}$ class of operators and thus it is redundant and
can not be a part of the operator basis.

Alternatively, using the notion of integration by parts (IBP) we have the
following relation:

$\displaystyle(\mathcal{D}_{\mu}\,X^{\mu\nu})^{2},\hskip
2.84544pt(\mathcal{D}_{\mu}\,X^{\mu\nu})(\mathcal{D}_{\mu}\,\tilde{X}^{\mu\nu})\,\,$
$\displaystyle\xRightarrow{\text{IBP}}$
$\displaystyle[\mathcal{D}_{\mu},\,\mathcal{D}_{\nu}]X^{[\mu\kappa}\,X_{\kappa}^{\nu]},\hskip
2.84544pt[\mathcal{D}_{\mu},\,\mathcal{D}_{\nu}]X^{[\mu\kappa}\,\tilde{X}_{\kappa}^{\nu]}$
(24) $\displaystyle\equiv$ $\displaystyle
X_{\mu\nu}X^{\mu\kappa}X_{\kappa}^{\nu},\hskip
2.84544ptX_{\mu\nu}X^{\mu\kappa}\tilde{X}_{\kappa}^{\nu}.$

Here, $[\mathcal{D}_{\mu},\,\mathcal{D}_{\nu}]$ is suitably replaced by
$X_{\mu\nu}$ and we have obtained $X^{3}$ class of operators. So, we conclude
that with the help of EOMs and IBPs, the operators belonging to
$\mathcal{D}^{2}\,X^{2}_{L,R}$ and $\mathcal{D}^{2}\,X_{L}\,X_{R}$ classes can
always be recast into operators of other classes. Thus these two are excluded
from the operator basis.

  * •

$\boxed{\Phi^{2}\,X_{L,R}\,\mathcal{D}^{2}}$ : The covariant form of this
operator
$(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\nu}\,\Phi)\,\mathcal{D}_{\mu}\,X^{\mu\nu}$
can be rewritten using Eqn. (21) as:

$\displaystyle(\Phi^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\nu}\,\Phi)\,\mathcal{D}_{\mu}\,X^{\mu\nu}=a^{\prime}\,(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})(\Phi^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{\nu}\,\Phi)+b^{\prime}\,(\Phi^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\nu}\,\Phi)(\Phi^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{\nu}\,\Phi),$
(25)

where $a^{\prime},b^{\prime}$ are complex numbers. Similar to the previous
case, $\Phi^{2}X\,\mathcal{D}^{2}$ can be rewritten in terms of operator
classes $\Psi_{L}\Psi_{R}\,\Phi^{2}\mathcal{D}$ and $\Phi^{4}\mathcal{D}^{2}$.
This justifies the absence of $\Phi^{2}\,X_{L,R}\,\mathcal{D}^{2}$ class from
the independent operator set.

  * •

$\boxed{\Psi_{L}\,\Psi_{R}\,X_{L,R}\,\mathcal{D}}$ : We find two different
covariant forms
$X^{\mu\nu}\,(\overline{\Psi}_{L,R}\,\gamma_{\mu}\,\mathcal{D}_{\nu}\,\Psi_{L,R})$
and
$(\mathcal{D}_{\mu}\,X^{\mu\nu})(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})$.
These operators can be further reduced with the help of suitable EOMs as:

$\displaystyle
X^{\mu\nu}\,(\overline{\Psi}_{L,R}\,\gamma_{\mu}\,\mathcal{D}_{\nu}\,\Psi_{L,R})$
$\displaystyle=$ $\displaystyle
X^{\mu\nu}\,(\overline{\Psi}_{L,R}\,\gamma_{\mu}\,\gamma_{\nu}\,\not{\mathcal{D}}\,\Psi_{L,R})=X^{\mu\nu}\,(\overline{\Psi}_{L,R}\,\gamma_{[\mu}\,\gamma_{\nu]}\,\not{\mathcal{D}}\,\Psi_{L,R})$
(26) $\displaystyle=$ $\displaystyle
X^{\mu\nu}\,(\overline{\Psi}_{L,R}\,\sigma_{\mu\nu}\,\Psi_{R,L})\,\Phi\,\,\,\,\,\equiv\,\,\,\Psi^{2}\,\Phi\,X,$
$\displaystyle(\mathcal{D}_{\mu}\,X^{\mu\nu})(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})$
$\displaystyle=$ $\displaystyle
c^{\prime}_{1}\,(\overline{\Psi}\,\gamma^{\nu}\,\Psi)\,(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})+c^{\prime}_{2}\,(\Phi^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{\nu}\,\Phi)\,(\overline{\Psi}_{L,R}\,\gamma_{\nu}\,\Psi_{L,R})$
(27) $\displaystyle\equiv$
$\displaystyle\Psi^{4}_{L,R}\,/\,\Psi^{2}_{L}\,\Psi^{2}_{R}+\Psi_{L}\,\Psi_{R}\,\Phi^{2}\,\mathcal{D}.$

Thus it is quite evident why this class is also counted as redundant.

In summary, the symmetries of the theory play a crucial role in constructing
the invariant operator set. But it is not guaranteed that all of them are
independent and thus the set of operators is always over-complete. To be a
part of the Lagrangian the operators of any mass dimension must form a basis,
i.e., the operators should be independent. To ensure that we have shown
through some toy examples how the EOMs and IBPs relate different operators and
thus can be used as constraints in this computation. In the latter part of
this paper, we have computed the dimension 6 operator basis for a plethora of
models. As the “Warsaw” is the only known complete operator basis, we have
tabulated our results in this basis only. There is another popular choice -
the SILH (Strongly Interacting Light Higgs) basis which trades away the
fermion rich operator classes $\Psi^{4},\,\Psi^{2}\Phi
X,\,\Psi^{2}\Phi^{2}\mathcal{D}$ from the Warsaw one and includes
$\mathcal{D}^{2}X^{2},\,\Phi^{2}X\mathcal{D}^{2}$, see Fig. 1.

Figure 1: All possible Lorentz invariant dimension 6 operator classes shown as
part of the Warsaw and SILH bases for SMEFT. The arrows depict relations among
the classes based on the equations of motion (EOMs) of various fields.

####  Symmetry generators and their identities

The quantum fields that are the building blocks of the operators transform
under the assigned space-time and internal symmetries. The symmetry generators
(specifically for the non-abelian case) respect the pre-fixed algebras and
satisfy a few identities. For example, the Lorentz symmetry generators
$\sigma^{\mu},\overline{\sigma}^{\mu}$ together form the
$\sigma^{\mu\nu},\overline{\sigma}^{\mu\nu}$ matrices, defined in Eqn. (28):

$\displaystyle(\sigma^{\mu\nu})_{\alpha}^{\beta}=(\sigma^{\mu})_{\alpha\dot{\beta}}(\overline{\sigma}^{\nu})^{\dot{\beta}\beta},\hskip
28.45274pt(\overline{\sigma}_{\mu\nu})_{\dot{\alpha}}^{\dot{\beta}}=(\overline{\sigma}_{\mu})^{\dot{\beta}\beta}(\sigma_{\nu})_{\beta\dot{\alpha}}\,\,.$
(28)

They also satisfy the following identities Dreiner:2008tw :

$\displaystyle(\sigma^{\mu})_{\alpha\dot{\alpha}}(\sigma_{\mu})_{\beta\dot{\beta}}$
$\displaystyle=$ $\displaystyle
2\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\,,$ (29)
$\displaystyle(\sigma^{\mu})_{\alpha\dot{\alpha}}(\overline{\sigma}_{\mu})^{\dot{\beta}\beta}$
$\displaystyle=$ $\displaystyle
2\delta^{\beta}_{\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}\,,$ (30)
$\displaystyle(\overline{\sigma}^{\mu})^{\dot{\alpha}\alpha}(\overline{\sigma}_{\mu})^{\dot{\beta}\beta}$
$\displaystyle=$ $\displaystyle
2\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}\,,$ (31)
$\displaystyle[\sigma^{\mu}\overline{\sigma}^{\nu}+\sigma^{\nu}\overline{\sigma}^{\mu}]_{\alpha}^{\beta}$
$\displaystyle=$ $\displaystyle 2g^{\mu\nu}\delta_{\alpha}^{\beta}\,.$ (32)

The internal symmetry generators respect their algebra as well as some related
identities. For example, the $SU(2)$ and $SU(3)$ generators, Pauli matrices
$\tau^{I}(I=1,2,3)$ and the Gell-Mann matrices $T^{A}(A=1,2,\cdots,8)$
respectively satisfy the following identities:

$\displaystyle\tau^{I}_{ij}\tau^{I}_{kl}=$ $\displaystyle
2\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}\,,$ (33) $\displaystyle
T^{A}_{ij}T^{A}_{kl}=$
$\displaystyle\frac{1}{2}\delta_{il}\delta_{jk}-\frac{1}{6}\delta_{ij}\delta_{kl}\,.$
(34)

While constructing the covariant form of the operators we may encounter two
different structures with the same field content. But they need not be two
independent operators and may be related to each other through these
identities Eqns. (29)-(34). Here, we have demonstrated how the utilisation of
these identities could help us to relate different covariant-structured
dimension 6 operators with a few examples777For this particular discussion we
have suppressed the chiral-indices..

  * •

$\boxed{\Psi^{4}}$: We have considered two dimension 6 operators
$(\overline{d}\,\gamma^{\mu}\,T^{A}\,d)(\overline{Q}\,\gamma_{\mu}\,T^{A}\,Q)$
and $(\overline{d}\,\gamma^{\mu}\,d)(\overline{Q}\,\gamma_{\mu}\,Q)$ from this
class. Using the identities in Eqns. (29)-(33), these operators can be
expressed as:

$\displaystyle(\overline{d}\gamma^{\mu}T^{A}d)(\overline{Q}T^{A}\gamma_{\mu}Q)$
$\displaystyle=(\overline{d}^{\alpha}\sigma^{\mu}_{\alpha\dot{\alpha}}T^{A}d^{\dot{\alpha}})(\overline{Q}_{\dot{\beta}}T^{A}\overline{\sigma}^{\mu\dot{\beta}\beta}Q_{\beta})=2\overline{d}^{\alpha}T^{A}Q_{\beta}\overline{Q}_{\dot{\beta}}T^{A}d^{\dot{\alpha}}\delta^{\beta}_{\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}$
(36)
$\displaystyle=2(\overline{d}\,T^{A}\,Q)\,(\overline{Q}\,T^{A}\,d)=2(\overline{d}_{a}\,[T^{A}]_{b}^{a}\,Q^{b})(\overline{Q}_{c}\,[T^{A}]_{e}^{c}\,d^{e})$
$\displaystyle=(\overline{d}\,d)\,(\overline{Q}\,Q)-\frac{1}{3}(\overline{d}\,Q)(\overline{Q}\,d)\,.$
$\displaystyle(\overline{d}\gamma^{\mu}d)(\overline{Q}\gamma_{\mu}Q)$
$\displaystyle=(\overline{d}^{\alpha}\sigma^{\mu}_{\alpha\dot{\alpha}}d^{\dot{\alpha}})(\overline{Q}_{\dot{\beta}}\overline{\sigma}^{\mu\dot{\beta}\beta}Q_{\beta})=2\overline{d}^{\alpha}Q_{\beta}\overline{Q}_{\dot{\beta}}d^{\dot{\alpha}}\delta^{\beta}_{\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}$
$\displaystyle=2(\overline{d}Q)(\overline{Q}d).$

It is quite evident from Eqns (36) and (36), that with the fields $d$,
$\bar{d}$, $Q$, and $\bar{Q}$ we can only have two independent operators that
should be included in SMEFT dimension 6 operator basis. Similarly, with fields
$e$, $\bar{L}$, $u$, and $\bar{Q}$ we have following relation

$\displaystyle(\bar{L}\sigma_{\mu\nu}e)(\bar{Q}\sigma^{\mu\nu}u)$
$\displaystyle=((\bar{L})^{\alpha}(\sigma_{\mu\nu})_{\alpha}^{\beta}(e)_{\beta})((\bar{Q})^{\rho}(\sigma^{\mu\nu})_{\rho}^{\theta}(u)_{\theta})$
(37)
$\displaystyle=((\bar{L})^{\alpha}(\sigma_{\mu})_{\alpha\dot{\beta}}(\overline{\sigma}_{\nu})^{\dot{\beta}\beta}(e)_{\beta})((\bar{L})^{\rho}(\sigma^{\mu})_{\rho\dot{\theta}}(\overline{\sigma}^{\nu})^{\dot{\theta}\theta}(u)_{\theta})$
$\displaystyle=4(\bar{L}e)(\bar{Q}u)-8(\bar{L}u)(\bar{Q}e)\,.$

and thus, only $(\bar{L}\sigma_{\mu\nu}e)(\bar{Q}\sigma^{\mu\nu}u)$ and
$(\bar{L}\,e)(\bar{Q}\,u)$ are included in the operator set.

  * •

$\boxed{\Phi^{6}}$ : Here, we are looking into the quartic subpart of the
dimension 6 operator $(H^{\dagger}\,H)^{3}$. It is interesting to note using
Eqn. (33) that inclusion of $SU(2)$ generators does not lead to an independent
operator in the SMEFT basis Grzadkowski:2010es :

$\displaystyle(H^{\dagger}\tau^{I}H)(H^{\dagger}\tau^{I}H)$
$\displaystyle=(H^{\dagger}_{i}\tau^{I}_{ij}H_{j})(H^{\dagger}_{k}\tau^{I}_{kl}H_{l})=H^{\dagger}_{i}H_{j}H^{\dagger}_{k}H_{l}(2\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl})$
(38)
$\displaystyle=2(H^{\dagger}H)^{2}-(H^{\dagger}H)^{2}=(H^{\dagger}H)^{2}\,.$

  * •

$\boxed{\Phi^{4}\mathcal{D}^{2}}$ : To illustrate the redundancy in this class
of operators, we have considered an operator involving a scalar Lepto-Quark
($\chi_{1}$) transforming as $(3,2,1/6)$ under the SM gauge group.

$\displaystyle(H^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}^{I}H)(\chi_{1}^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\mu
I}\chi_{1})$
$\displaystyle=(H^{\dagger}(\tau^{I}i\mathcal{D}_{\mu}-i\overleftarrow{\mathcal{D}}_{\mu}\tau^{I})H)(\chi^{\dagger}_{1}(\tau^{I}i\mathcal{D}^{\mu}-i\overleftarrow{\mathcal{D}}^{\mu}\tau^{I})\chi_{1})$
(39)
$\displaystyle=(H^{\dagger}_{i}\tau^{I}_{ij}(i\mathcal{D}_{\mu}H)_{j}-(i\mathcal{D}_{\mu}H)^{\dagger}_{k}\tau^{I}_{kl}H_{l})(\chi_{1a}^{\dagger}\tau^{I}_{ab}(i\mathcal{D}^{\mu}\chi_{1})_{b}-(i\mathcal{D}^{\mu}\chi_{1})^{\dagger}_{c}\tau^{I}_{cd}\chi_{1d})$
$\displaystyle=-(H^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}H)(\chi_{1}{{}^{\dagger}}i\overleftrightarrow{\mathcal{D}}^{\mu}\chi_{1})+2\,(H^{\dagger}_{b}i\overleftrightarrow{\mathcal{D}}_{\mu}H^{a})(\chi_{1a}^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\mu}\chi_{1}^{b}).$

As three operators are related through the above relation, only two of these
can be independent and we may include
$(H^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}^{I}H)(\chi_{1}^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\mu
I}\chi_{1})$ and
$(H^{\dagger}i\overleftrightarrow{\mathcal{D}}_{\mu}H)(\chi_{1}^{\dagger}i\overleftrightarrow{\mathcal{D}}^{\mu}\chi_{1})$
in the operator basis for this scenario.

  * •

$\boxed{\Psi^{2}\Phi^{2}\mathcal{D}}$ : In this class we can have following
three operators involving the Lepto-Quark $\chi_{1}$:

$\displaystyle(\overline{Q}\tau^{I}\gamma^{\mu}Q)\,(\chi^{\dagger}_{1}i\overleftrightarrow{\mathcal{D}}_{\mu}^{I}\chi_{1})$
$\displaystyle=$
$\displaystyle(\overline{Q}\tau^{I}\gamma^{\mu}Q)[\chi_{1}^{\dagger}\tau^{I}(i\mathcal{D}_{\mu}\chi_{1})+(i\mathcal{D}_{\mu}\chi_{1})^{\dagger}\tau^{I}\chi_{1}]$
(40) $\displaystyle=$ $\displaystyle
2(\overline{Q}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1}))(Q\chi_{1}^{\dagger})-(\overline{Q}\gamma^{\mu}Q)(\chi_{1}^{\dagger}(i\mathcal{D}_{\mu}\chi_{1}))$
$\displaystyle+2(\overline{Q}\gamma^{\mu}\chi_{1})(Q(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})-(\overline{Q}\gamma^{\mu}Q)((i\mathcal{D}_{\mu}\chi_{1})^{\dagger}\chi_{1}),$
$\displaystyle(\overline{Q}T^{A}\gamma^{\mu}\,Q)\,(\chi^{\dagger}_{1}i\overleftrightarrow{\mathcal{D}}_{\mu}^{A}\chi_{1})$
$\displaystyle=$
$\displaystyle(\overline{Q}T^{A}\gamma^{\mu}Q)[\chi_{1}^{\dagger}T^{A}(i\mathcal{D}_{\mu}\chi_{1})+(i\mathcal{D}_{\mu}\chi_{1})^{\dagger}T^{A}\chi_{1}]$
(41) $\displaystyle=$
$\displaystyle\frac{1}{2}(\overline{Q}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1}))(Q\chi_{1}^{\dagger})-\frac{1}{6}(\overline{Q}\gamma^{\mu}Q)(\chi_{1}^{\dagger}(i\mathcal{D}_{\mu}\chi_{1}))$
$\displaystyle+\frac{1}{2}(\overline{Q}\gamma^{\mu}\chi_{1})(Q(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})-\frac{1}{6}(\overline{Q}\gamma^{\mu}Q)((i\mathcal{D}_{\mu}\chi_{1})^{\dagger}\chi_{1}),$
$\displaystyle(\overline{Q}\,T^{A}\,\tau^{I}\,\gamma^{\mu}\,Q)\,(\chi^{\dagger}_{1}\,T^{A}\,i\overleftrightarrow{\mathcal{D}}_{\mu}^{I}\,\chi_{1})$
$\displaystyle=$
$\displaystyle(\overline{Q}T^{A}\tau^{I}\gamma^{\mu}Q)[\chi_{1}^{\dagger}T^{A}\tau^{I}(i\mathcal{D}_{\mu}\chi_{1})+(i\mathcal{D}_{\mu}\chi_{1})^{\dagger}T^{A}\tau^{I}\chi_{1}]$
(42) $\displaystyle=$
$\displaystyle\frac{1}{2}[(\overline{Q}\tau^{I}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1}))(Q\tau^{I}\chi_{1}^{\dagger})]-\frac{1}{6}[(\overline{Q}\tau^{I}\gamma^{\mu}Q)(\chi_{1}^{\dagger}\tau^{I}(i\mathcal{D}_{\mu}\chi_{1}))]$
$\displaystyle+\frac{1}{2}[(\overline{Q}\tau^{I}\gamma^{\mu}\chi_{1})(Q\tau^{I}(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})]-\frac{1}{6}[(\overline{Q}\tau^{I}\gamma^{\mu}Q)((i\mathcal{D}_{\mu}\chi_{1})^{\dagger}\tau^{I}\chi_{1})]$
$\displaystyle=$
$\displaystyle[(\overline{Q}\gamma^{\mu}\chi_{1}^{\dagger})((i\mathcal{D}_{\mu}\chi_{1})Q)-\frac{1}{2}((\overline{Q}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1}))(Q\chi_{1}^{\dagger}))]$
$\displaystyle-\frac{1}{6}[(\overline{Q}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1}))(Q\chi_{1}^{\dagger})-(\overline{Q}\gamma^{\mu}Q)(\chi_{1}^{\dagger}(i\mathcal{D}_{\mu}\chi_{1}))]$
$\displaystyle+[(\overline{Q}\gamma^{\mu}(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})(\chi_{1}Q)-\frac{1}{2}(\overline{Q}\gamma^{\mu}\chi_{1})(Q(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})]$
$\displaystyle-\frac{1}{6}[(\overline{Q}\gamma^{\mu}\chi_{1})(Q(i\mathcal{D}_{\mu}\chi_{1})^{\dagger})-(\overline{Q}\gamma^{\mu}Q)((i\mathcal{D}_{\mu}\chi_{1})^{\dagger}\chi_{1})].$

Thus, it is evident that the three operators in the LHS of the above equation
along with
$(\overline{Q}\gamma^{\mu}Q)\,(\chi^{\dagger}_{1}i\overleftrightarrow{\mathcal{D}}_{\mu}\chi_{1})$,
comprise a set of four independent operators and qualify to be in the operator
basis.

###  2.4 Additional impacts of the Global (Accidental) Symmetries

The effect of global symmetries is very similar to the gauge ones in the
construction of invariant operators. But, unlike the gauge symmetry, the
global symmetry need not be strictly imposed and it may be allowed to be
broken softly in specific interactions as demanded by the phenomenology. This
leads to the appearance of global charge violating effective operators that
induce rare processes.

Baryon ($B$) and lepton ($L$) numbers appear as accidental global symmetries
in the tree-level SM Lagrangian, see Eqn. (46). But they may be violated
through higher dimensional operators. If we assign the leptons an $L$ charge
of -1 unit and the quarks a $B$ charge of 1/3 units respectively, then we can
generate a Majorana neutrino mass for the SM neutrinos through dimension 5
$H^{2}L^{2}$ operator. As this operator is suppressed by a high scale, the
smallness of neutrino masses can be explained. Similarly within the SMEFT
framework, we find operators violating $B$ and $L$ by $(0,-2)$, $(1,-1)$,
$(1,1)$ units at mass dimensions 5, 6 and 7. Recently it has been noted
Hambye:2013jsa  that a similar violation by $(1,-3)$ units appears at
dimension 9 and this can induce a new decay mode of the proton to three
charged leptons.

In the case of BSM scenarios, there could be additional global symmetries and
the amount of their breaking would be completely phenomenologically driven.
This controls the appearance of certain kinds of operators at different mass
dimensions.

##  3 BSMEFT Operator Bases

The spectrum of the UV complete theory that is expected to explain all
shortcomings of the SM is non-degenerate. This implies the existence of a
multitude of scales associated with BSM fields of different masses. Thus even
if all the non-SM particles are integrated out, all the higher dimensional
operators will not be suppressed by a single cut-off scale ($\Lambda$). The
natural scenario would be the presence of a tower of effective operators
involving different $\Lambda$’s lying between the electroweak and the unknown
UV scales. Unless the BSM spectrum is really compressed, the lightest non-SM
particle is expected to be within the reach of the ongoing experiments
($\leq\mathcal{O}$(TeV)) where the rest of the new particles are heavy enough
to be successfully integrated out. In this framework, that lightest non-SM
particle should be treated as an IR-DOF along with the SM ones and we must
compute the effective operators involving them to capture the effects of the
full UV theory. This has been the motivation of our BSMEFT construction. The
most generic choices for non-SM IR-DOFs are real and complex scalar and
fermion multiplets, vector like fermions, and Lepto-Quark bosons under the SM
gauge symmetry. There may be additional gauge bosons as well. The choice of
these fields is motivated from the fact that most of the phenomenologically
interesting scenarios contain these DOFs in their (non)minimal versions. We
have schematically demonstrated the idea using some example scenarios in Fig.
2 where it is quite evident that there could be multiple parent UV theories
which may lead to the same set of lighter particles. Thus one BSMEFT operator
basis qualifies to encapsulate the features of all such UV theories treating
them degenerate. To discriminate between them, we need to identify the subset
of that BSMEFT operator basis corresponding to each of the UV theories. This
is beyond the goal of this paper and will be discussed in our upcoming
article.

Based on the previous discussion, we have considered three different
extensions of the SM and for each such scenario we have included multiple
examples to encompass the most popular choices. For each example model, we
have constructed the complete and independent BSMEFT operator bases up to mass
dimension 6. Here, we have tabulated only the additional effective operators
beyond SMEFT. For the sake of completeness the SMEFT dimension 6 operators are
noted in the appendix.

Figure 2: An elucidation of the inter-connectedness of several BSM scenarios:
paving the path to BSMEFT.

###  3.1 Standard Model extended by uncolored particles

To start with, we have considered the scenarios where the SM is extended by
suitable addition of extra uncolored particle(s), e.g., $SU(2)_{L}$ complex
singlets and higher multiplets with fermionic and bosonic degrees of freedom.
These particles can be part of an $SU(2)_{L}$ multiplet as well. The
electromagnetic charges of the singlet fields are solely determined by their
assigned hypercharges888Our working formula is $Q=T_{3}+Y$ where $Q,T_{3},Y$
are electromagnetic charge, 3rd component of isospin and hypercharge
respectively.. We have summarized the quantum numbers of the non-SM fields in
Table 2.

Model No. | Non-SM IR DOFs | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | Spin
---|---|---|---|---|---
(Color Singlets)
1 | $\delta^{+}$ | 1 | 1 | 1 | 0
2 | $\rho^{++}$ | 1 | 1 | 2 | 0
3 | $\Delta$ | 1 | 3 | 1 | 0
4 | $\Sigma$ | 1 | 3 | 0 | 1/2
5 | $V_{L,R}$ | 1 | 2 | -1/2 | 1/2
$E_{L,R}$ | 1 | 1 | -1 | 1/2
$N_{L,R}$ | 1 | 1 | 0 | 1/2
Table 2: Additional IR DOFs (Color Singlets) as representations of the SM
gauge groups along with their spin quantum numbers.

#### SM + Singly Charged Scalar ($\delta^{+}$)

We have considered the extension of SM by an $SU(2)_{L}$ singlet complex
scalar field ($\delta^{+}$) of hypercharge 1, see Table 2. After the
spontaneous electroweak symmetry breaking this field emerges as a singly
charged physical scalar field999Although these particles are added to the
unbroken SM gauge symmetry, but looking into this feature we will identify
this and other fields by their electromagnetic charges.. It is interesting to
note that when the SM is embedded in an extended gauge symmetry, e.g., Left
Right Symmetric Model (LRSM), then the appearance of singly charged scalar(s)
is unavoidable once the additional symmetry is broken to the SM. There are
attempts to generate neutrino masses either radiatively or through higher
dimensional operators where the SM is extended by mutiple $SU(2)$ singlet
complex scalars, e.g., see Ref. Cheng:1980qt ; Babu:1988ki ; Babu:2002uu ;
AristizabalSierra:2006gb . This motivates us to construct an effective theory
with this simplest non-trivial extension of the SM. We have categorized the
effective operators involving $\delta^{+}$ of dimensions 5 and 6 in Tables 3
and 4. The operators with distinct hermitian conjugates have been coloured
blue.

Features of the additional operators:

  * •

Here, we have noted two types of dimension 5 operators - i) $B,L$ conserving
$\tilde{\mathcal{O}}_{QdH\delta}$, $\tilde{\mathcal{O}}_{uQH\delta}$,
$\tilde{\mathcal{O}}_{LeH\delta}$, and ii) $L$ violating
$\tilde{\mathcal{O}}_{e\delta}$ which are highlighted in red colour in Table
3.

  * •

The additional dimension 6 operators of class $\Phi^{6}$ and
$\Phi^{4}\mathcal{D}^{2}$ mimic their SM counterparts.

  * •

Since, $\delta^{+}$ is an $SU(2)_{L}$ singlet there is no mixing between
$B_{\mu\nu}$ and $W^{I}_{\mu\nu}$ in the $\Phi^{2}X^{2}$ class nor do we
obtain higher tensor products in the $\Psi^{2}\Phi^{2}\mathcal{D}$ class.

  * •

The operators, highlighted in red colour in Table 4, violate lepton number by
two units in the $\Psi^{2}\Phi^{2}\mathcal{D}$, $\Psi^{2}\Phi^{3}$ and
$\Psi^{2}\Phi X$ classes.

$\Psi^{2}\Phi^{2}$
---
$\tilde{\mathcal{O}}_{QdH\delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N^{2}_{f})\,\epsilon_{ij}\,(\overline{Q}_{p\alpha i}\,d_{q}^{\alpha})\,(\tilde{H}_{j}\,\delta)}$ | $\tilde{\mathcal{O}}_{uQH\delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N^{2}_{f})\,(\overline{u}_{p\alpha}\,Q^{\alpha i}_{q})\,(\tilde{H}_{i}\,\delta)}$
$\tilde{\mathcal{O}}_{LeH\delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N^{2}_{f})\,\epsilon_{ij}\,(\overline{L}_{pi}\,e_{q})\,(\tilde{H}_{j}\,\delta)}$ | $\tilde{\mathcal{O}}_{e\delta}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N^{2}_{f}+N_{f})\,(e_{p}^{T}\,C\,e_{q})\,\delta^{2}}$
Table 3: SM extended by Singly Charged Scalar ($\delta$): Additional operators of dimension 5. $\delta$, $\delta^{\dagger}$ represent $\delta^{+}$ and $\delta^{-}$ respectively. Here $i,j$ and $\alpha$ are the $SU(2)$ and $SU(3)$ indices respectively. $p,q=1,2,\cdots,N_{f}$ are the flavour indices. The operator in red violates lepton number. $\Phi^{6}$ | $\Phi^{4}\mathcal{D}^{2}$
---|---
$\mathcal{O}_{\delta}$ | $(\delta^{\dagger}\,\delta)^{3}$ | $\mathcal{O}_{\delta\square}$ | $(\delta^{\dagger}\,\delta)\,\square\,(\delta^{\dagger}\,\delta)$
$\mathcal{O}_{H^{2}\delta^{4}}$ | $(H^{\dagger}\,H)\,(\delta^{\dagger}\,\delta)^{2}$ | $\mathcal{O}_{H\delta\mathcal{D}}^{(1)}$ | $(\delta^{\dagger}\,\delta)\,\left[(\mathcal{D}^{\mu}\,H)^{\dagger}(\mathcal{D}_{\mu}\,H)\right]$
$\mathcal{O}_{H^{4}\delta^{2}}$ | $(H^{\dagger}\,H)^{2}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{H\delta\mathcal{D}}^{(2)}$ | $(H^{\dagger}\,H)\,\left[(\mathcal{D}^{\mu}\,\delta)^{\dagger}(\mathcal{D}_{\mu}\,\delta)\right]$
$\Phi^{2}X^{2}$ | $\Psi^{2}\Phi^{2}\mathcal{D}$
$\mathcal{O}_{B\delta}$ | $B_{\mu\nu}\,B^{\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{Q\delta\mathcal{D}}$ | $(N^{2}_{f})(\,\overline{Q}_{p\alpha i}\,\gamma^{\mu}\,Q^{\alpha i}_{q})\,(\delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\delta)$
$\mathcal{O}_{\tilde{B}\delta}$ | $\tilde{B}_{\mu\nu}\,B^{\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{L\delta\mathcal{D}}$ | $(N^{2}_{f})(\,\overline{L}_{pi}\,\gamma^{\mu}\,L^{i}_{q})\,(\delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\delta)$
$\mathcal{O}_{G\delta}$ | $G^{A}_{\mu\nu}\,G^{A\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{u\delta\mathcal{D}}$ | $(N^{2}_{f})(\,\overline{u}_{p\alpha}\,\gamma^{\mu}\,u^{\alpha}_{q})\,(\delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\delta)$
$\mathcal{O}_{\tilde{G}\delta}$ | $\tilde{G}^{A}_{\mu\nu}\,G^{A\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{d\delta\mathcal{D}}$ | $(N^{2}_{f})(\,\overline{d}_{p\alpha}\,\gamma^{\mu}\,d^{\alpha}_{q})\,(\delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\delta)$
$\mathcal{O}_{W\delta}$ | $W^{I}_{\mu\nu}\,W^{I\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{e\delta\mathcal{D}}$ | $(N^{2}_{f})(\,\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\delta)$
$\mathcal{O}_{\tilde{W}\delta}$ | $\tilde{W}^{I}_{\mu\nu}\,W^{I\mu\nu}\,(\delta^{\dagger}\,\delta)$ | $\mathcal{O}_{LeH\delta\mathcal{D}}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N^{2}_{f})\,((L_{p}^{i})^{T}\,\gamma^{\mu}\,e_{q})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\delta)}$
$\Psi^{2}\Phi^{3}$
$\mathcal{O}_{LeH\delta}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N^{2}_{f})\,(\overline{L}_{pi}\,e_{q})\,H^{i}\,(\delta^{\dagger}\,\delta)}$ | $\mathcal{O}_{LH\delta}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N^{2}_{f})\,\epsilon_{ij}\,((L_{p}^{i})^{T}\,C\,L_{q}^{k})\,\delta\,(H^{\dagger}_{k}\,H^{j})}$
$\mathcal{O}_{QuH\delta}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N^{2}_{f})\,\epsilon_{ij}\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q})\,\tilde{H}_{j}\,(\delta^{\dagger}\,\delta)}$ | $\mathcal{O}_{L\delta}$ | $\boxed{{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}-N_{f})\,\epsilon_{ij}\,((L_{p}^{i})^{T}\,C\,L_{q}^{j})\,\delta\,(\delta^{\dagger}\,\delta)}}$
$\mathcal{O}_{QdH\delta}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N^{2}_{f})\,(\overline{Q}_{p\alpha i}\,d^{\alpha}_{q})\,H^{i}\,(\delta^{\dagger}\,\delta)}$ |  |
$\Psi^{2}\Phi X$
$\mathcal{O}_{BL\delta}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}+N_{f})\,\epsilon_{ij}\,B_{\mu\nu}\,((L^{i}_{p})^{T}\,C\,\sigma^{\mu\nu}\,L^{j}_{q})\,\delta}$ | $\mathcal{O}_{WL\delta}$ | $\boxed{{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}-N_{f})\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,((L^{i}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\tau^{I}\,L^{j}_{q})\,\delta}}$
Table 4: SM extended by Singly Charged Scalar ($\delta$): Additional operators
of dimension 6. Boxed operators vanish for single flavour. $\delta$,
$\delta^{\dagger}$ represent $\delta^{+}$ and $\delta^{-}$ respectively. Here
$i,j$ and $\alpha$ are the $SU(2)$ and $SU(3)$ indices respectively.
$\tau^{I}$ is $SU(2)$ generator. $A=1,2,\cdots,8$ and $I=1,2,3.$
$p,q=1,2,\cdots,N_{f}$ are the flavour indices. Operators in red violate
lepton number.

#### SM + Doubly Charged Scalar ($\rho^{++}$)

Similar to the earlier case, when the SM is emerged from LRSM gauge theory,
the right handed complex triplet may lead to an additional scalar of
hypercharge 2 which is further identified as a doubly charged scalar
($\rho^{++}$), see Table 2. Also scenarios like in Ref. Cheng:1980qt ;
Babu:1988ki ; Rizzo:1981xx ; Akeroyd:2005gt ; Maalampi:2002vx ; Babu:2002uu ;
AristizabalSierra:2006gb  contain a single doubly charged scalar. Here, our
primary concern is to construct the effective operators involving the
additional doubly charged scalar and thus mimic the concept of Refs.
Chakrabortty:2015zpm ; King:2014uha . We have provided the effective operators
involving $\rho^{++}$ up to dimension 6 in Table 5. Operators with distinct
hermitian conjugates have been coloured blue.

$\Phi^{6}$ | $\Phi^{4}\mathcal{D}^{2}$
---|---
$\mathcal{O}_{\rho}$ | $(\rho^{\dagger}\,\rho)^{3}$ | $\mathcal{O}_{\rho\square}$ | $(\rho^{\dagger}\,\rho)\,\square\,(\rho^{\dagger}\,\rho)$
$\mathcal{O}_{H^{2}\rho^{4}}$ | $(H^{\dagger}\,H)\,(\rho^{\dagger}\,\rho)^{2}$ | $\mathcal{O}_{H\rho\mathcal{D}}^{(1)}$ | $(\rho^{\dagger}\,\rho)\,\left[(\mathcal{D}^{\mu}\,H)^{\dagger}(\mathcal{D}_{\mu}\,H)\right]$
$\mathcal{O}_{H^{4}\rho^{2}}$ | $(H^{\dagger}\,H)^{2}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{H\rho\mathcal{D}}^{(2)}$ | $(H^{\dagger}\,H)\,\left[(\mathcal{D}^{\mu}\,\rho)^{\dagger}(\mathcal{D}_{\mu}\,\rho)\right]$
$\Phi^{2}X^{2}$ | $\Psi^{2}\Phi^{2}\mathcal{D}$
$\mathcal{O}_{B\rho}$ | $B_{\mu\nu}\,B^{\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{Q\rho\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,\gamma^{\mu}\,Q^{\alpha i}_{q})\,(\rho^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\rho)$
$\mathcal{O}_{\tilde{B}\rho}$ | $\tilde{B}_{\mu\nu}\,B^{\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{L\rho\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L^{i}_{q})\,(\rho^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\rho)$
$\mathcal{O}_{G\rho}$ | $G^{A}_{\mu\nu}\,G^{A\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{u\rho\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u^{\alpha}_{q})\,(\rho^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\rho)$
$\mathcal{O}_{\tilde{G}\rho}$ | $\tilde{G}^{A}_{\mu\nu}\,G^{A\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{d\rho\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d^{\alpha}_{q})\,(\rho^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\rho)$
$\mathcal{O}_{W\rho}$ | $W^{I}_{\mu\nu}\,W^{I\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{e\rho\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\rho^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\rho)$
$\mathcal{O}_{\tilde{W}\rho}$ | $\tilde{W}^{I}_{\mu\nu}\,W^{I\mu\nu}\,(\rho^{\dagger}\,\rho)$ | $\mathcal{O}_{LeH\rho\mathcal{D}}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N^{2}_{f})\,((L^{i}_{p})^{T}\,\gamma^{\mu}\,e_{q})\,(\,H^{\dagger}_{i}\,i\mathcal{D}_{\mu}\rho)}$
$\Psi^{2}\Phi^{3}$
$\mathcal{O}_{LeH\rho}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{L}_{pi}\,e_{q})\,H^{i}\,(\rho^{\dagger}\,\rho)}$ | $\mathcal{O}_{LH\rho}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}+N_{f})\,((L^{i}_{p})^{T}\,C\,L^{j}_{q})\,\rho\,(\tilde{H}_{i}\,\tilde{H}_{j})}$
$\mathcal{O}_{QuH\rho}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q})\,\tilde{H}^{j}\,(\rho^{\dagger}\,\rho)}$ | $\mathcal{O}_{e\rho}$ | ${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}+N_{f})\,(e_{p}^{T}\,C\,e_{q})\,\rho\,(\rho^{\dagger}\,\rho)}$
$\mathcal{O}_{QdH\rho}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,d^{\alpha}_{q})\,H^{i}\,(\rho^{\dagger}\,\rho)}$ | $\mathcal{O}_{eH\rho}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N^{2}_{f}+N_{f})\,(e_{p}^{T}\,C\,e_{q})\,\rho\,(H^{\dagger}\,H)}$
$\Psi^{2}\Phi X$
$\mathcal{O}_{Be\rho}$ | $\boxed{{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}(N^{2}_{f}-N_{f})\,B_{\mu\nu}\,(e_{p}^{T}\,C\,\sigma^{\mu\nu}\,e_{q})\,\rho}}$
Table 5: SM extended by Doubly Charged Scalar ($\rho$): Additional operators
of dimension 6. Boxed operators vanish for single flavour. $\rho$,
$\rho^{\dagger}$ represent $\rho^{++}$ and $\rho^{--}$ respectively. Here
$i,j$ and $\alpha$ are the $SU(2)$ and $SU(3)$ indices respectively.
$A=1,2,\cdots,8$ and $I=1,2,3.$ $p,q=1,2,\cdots,N_{f}$ are the flavour
indices. Operators in red violate lepton number.

Features of the additional operators:

  * •

One of the differences between the operator sets containing $\delta^{+}$ and
$\rho^{++}$ is the absence of dimension 5 operators in the latter case.

  * •

Similar to the earlier case, at dimension 6 there is no mixing between
$B_{\mu\nu}$ and $W_{\mu\nu}^{I}$ within the $\Phi^{2}X^{2}$ class and there
are no higher tensor products in the $\Psi^{2}\Phi^{2}\mathcal{D}$ class, on
account of $\rho^{++}$ being an $SU(2)_{L}$ singlet.

  * •

We have found new operators that violate lepton number by two units in the
$\Psi^{2}\Phi^{2}\mathcal{D}$, $\Psi^{2}\Phi^{3}$ and $\Psi^{2}\Phi X$
classes. These operators have been highlighted in red colour in Table 5.

#### SM + Complex Triplet Scalar ($\Delta$)

Here, we have explored other possible scenarios where electroweak multiplets
are assumed to be the lighter DOFs. First, we have considered a complex
$SU(2)_{L}$ triplet scalar ($\Delta$) having hypercharge of +1, Table 2. After
the spontaneous breaking of electroweak symmetry its components can be
assigned definite electromagnetic charges101010An $SU(2)$ triplet has $T_{3}$
values (+1, 0, -1). So, using $Q=T_{3}+Y$, we obtain the electromagnetic
charges (+2, +1, 0) since $Y=1$.
$(\Delta^{++},\,\,\Delta^{+},\,\,\Delta^{0})$. The complex triplet is
instrumental in mediating lepton number and flavour violating processes
delAguila:2013yaa ; delAguila:2013hla ; Fonseca:2018aav , interesting collider
signatures Chakrabortty:2015zpm , and also facilitates the generation of
neutrino mass Cheng:1980qt ; Konetschny:1977bn ; Schechter:1980gr ; Gu_2006 ;
Cai:2017mow . These observables may get affected by the interactions between
the heavier particles and this complex triplet, which can be captured through
the effective operators involving $\Delta$. A complex $SU(2)_{L}$ triplet can
descend from an LRSM once it is spontaneously broken to the SM, see Fig. 2.
There are many phenomenological models Georgi:1985nv ; Chiang:2015kka ;
Logan:2015xpa ; Chiang:2018cgb ; Chiang:2015amq ; Chiang:2012cn  where the SM
is extended by a complex triplet accompanied by multiple scalars and fermions.
In that case if the other particles are sufficiently heavier than the
$\Delta$, then they can be integrated out leading to an effective Lagrangian
with IR DOFs as SM ones and the complex triplet.

Here, We have listed the complete set of effective operators involving
$\Delta$, see Tables 6 and 7 for dimension 5 and 6 respectively. The operators
with distinct hermitian conjugates have been coloured blue. While writing the
operators, $\Delta$ has been expressed as a $2\times 2$ matrix
$\Delta^{I}\cdot\tau^{I}$ with $I$ = $1,2,3$ and $\tau^{I}$ being the Pauli
matrices.

$\Psi^{2}\Phi^{2}$ | $\Phi^{5}$
---|---
$\tilde{\mathcal{O}}_{LeH\Delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{N_{f}^{2}\,(\overline{L}_{pi}\,e_{q}\,\Delta\,\tilde{H}_{i})}$ | $\tilde{\mathcal{O}}_{H^{2}\Delta^{3}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(H^{T}\Delta^{\dagger}\,H)\,Tr[(\Delta^{\dagger}\,\Delta)]}$
$\tilde{\mathcal{O}}_{QdH\Delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{N_{f}^{2}\,(\overline{Q}_{p\alpha i}\,d^{\alpha}_{q}\,\Delta\,\tilde{H}_{i})}$ | $\tilde{\mathcal{O}}_{H^{2}\Delta^{3}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(H^{T}\Delta^{\dagger}\Delta^{\dagger}\,\Delta\,H)}$
$\tilde{\mathcal{O}}_{QuH\Delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{N_{f}^{2}\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q}\,\Delta^{\dagger}\,\tilde{H}_{i})}$ | $\tilde{\mathcal{O}}_{H^{4}\Delta}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(H^{T}\,\Delta^{\dagger}\,H)\,(H^{\dagger}\,H)}$
$\tilde{\mathcal{O}}_{e\Delta}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}+N_{f})\,(e_{p}^{T}\,C\,e_{q})\,Tr[\,\Delta\,\Delta]}$ |  |
Table 6: SM extended by Complex Triplet Scalar ($\Delta$): Additional
operators of dimension 5. Here $i$ and $\alpha$ are $SU(2)$ and $SU(3)$
indices respectively. $p,q=1,2,\cdots,N_{f}$ are the flavour indices. The
operator in red violates lepton number.

Features of the additional operators:

  * •

Contrary to $\delta^{+}$ and $\rho^{++}$, $\Delta$ transforms as an
$SU(2)_{L}$ triplet. This offers multiple ways to contract its indices to form
invariant operators, e.g., within
$\mathcal{O}^{(1),(2),(3)}_{H^{2}\Delta^{4}}$ class we have noted the
following partitions:

$\displaystyle\mathcal{O}^{(1)}_{H^{2}\Delta^{4}}\,\,$ $\displaystyle\equiv$
$\displaystyle\,\,H^{\dagger}\,(\Delta^{\dagger}\,\Delta)\,(\Delta^{\dagger}\,\Delta)\,H\hskip
25.6073pt\rightarrow(2\otimes 3\otimes 3\otimes 3\otimes 3\otimes 2),$
$\displaystyle\mathcal{O}^{(2)}_{H^{2}\Delta^{4}}\,\,$ $\displaystyle\equiv$
$\displaystyle\,\,Tr[(\Delta^{\dagger}\,\Delta)\,(\Delta^{\dagger}\,\Delta)]\,(H^{\dagger}\,H)\rightarrow(3\otimes
3\otimes 3\otimes 3)\otimes(2\otimes 2),$
$\displaystyle\mathcal{O}^{(3)}_{H^{2}\Delta^{4}}\,\,$ $\displaystyle\equiv$
$\displaystyle\,\,Tr[(\Delta^{\dagger}\,\Delta)]\,(H^{\dagger}\,\Delta^{\dagger}\,\Delta\,H)\hskip
7.11317pt\rightarrow(3\otimes 3)\otimes(2\otimes 3\otimes 3\otimes 2).$

Here, we pick a singlet representation from the tensor product within a
parenthesis.

$\Phi^{6}$ | $\Phi^{4}\mathcal{D}^{2}$
---|---
$\mathcal{O}_{\Delta}^{(1)}$ | $Tr[(\Delta^{\dagger}\,\Delta)]^{3}$ | $\mathcal{O}_{\Delta\square}$ | $Tr[(\Delta^{\dagger}\,\Delta)\,\square\,(\Delta^{\dagger}\,\Delta)]$
$\mathcal{O}_{\Delta}^{(2)}$ | $Tr[(\Delta^{\dagger}\,\Delta)\,(\Delta^{\dagger}\,\Delta)]\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{\Delta\mathcal{D}}^{(1)}$ | $Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}_{\mu}}\,\Delta)(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{\mu}\,\Delta)]$
$\mathcal{O}_{H^{2}\Delta^{4}}^{(1)}$ | $H^{\dagger}\,(\Delta^{\dagger}\,\Delta)\,(\Delta^{\dagger}\,\Delta)\,H$ | $\mathcal{O}_{\Delta\mathcal{D}}^{(2)}$ | $Tr[(\Delta^{\dagger}\,\Delta)]\,Tr[(\mathcal{D}^{\mu}\,\Delta^{\dagger})\,(\mathcal{D}_{\mu}\Delta)]$
$\mathcal{O}_{H^{2}\Delta^{4}}^{(2)}$ | $Tr[(\Delta^{\dagger}\,\Delta)\,(\Delta^{\dagger}\,\Delta)]\,(H^{\dagger}\,H)$ | $\mathcal{O}_{H\Delta\mathcal{D}}^{(1)}$ | $[H^{\dagger}\,(\mathcal{D}_{\mu}\Delta)]\,[(\mathcal{D}^{\mu}\,\Delta)^{\dagger}\,H]$
$\mathcal{O}_{H^{2}\Delta^{4}}^{(3)}$ | $Tr[(\Delta^{\dagger}\,\Delta)]\,(H^{\dagger}\,\Delta^{\dagger}\,\Delta\,H)$ | $\mathcal{O}_{H\Delta\mathcal{D}}^{(2)}$ | $[(\mathcal{D}^{\mu}\,H)^{\dagger}\,\Delta]\,[\Delta^{\dagger}\,(\mathcal{D}_{\mu}H)]$
$\mathcal{O}_{H^{4}\Delta^{2}}^{(1)}$ | $(H^{\dagger}\,\Delta^{\dagger}\,H)\,(H^{\dagger}\,\Delta\,H)$ | $\mathcal{O}_{H\Delta\mathcal{D}}^{(3)}$ | $Tr[(\Delta^{\dagger}\,\Delta)]\,(\mathcal{D}^{\mu}H)^{\dagger}\,(\mathcal{D}_{\mu}H)$
$\mathcal{O}_{H^{4}\Delta^{2}}^{(2)}$ | $(H^{\dagger}\,\Delta^{\dagger}\,\Delta\,\,H)\,(H^{\dagger}\,H)$ | $\mathcal{O}_{H\Delta\mathcal{D}}^{(4)}$ | $(H^{\dagger}\,H)\,Tr[(\mathcal{D}^{\mu}\Delta)^{\dagger}\,(\mathcal{D}_{\mu}\Delta)]$
$\mathcal{O}_{H^{4}\Delta^{2}}^{(3)}$ | $Tr[(\Delta^{\dagger}\,\Delta)]\,(H^{\dagger}\,H)^{2}$ |  |
$\mathcal{O}_{H\Delta H}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(H^{T}\,\Delta^{\dagger}\,H)^{2}}$ |  |
$\Phi^{2}X^{2}$ | $\Psi^{2}\Phi^{2}\mathcal{D}$
$\mathcal{O}_{B\Delta}$ | $B_{\mu\nu}\,B^{\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{Q\Delta\mathcal{D}}^{(1)}$ | $(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,\gamma^{\mu}\,Q^{\alpha i}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Delta)]$
$\mathcal{O}_{\tilde{B}\Delta}$ | $\tilde{B}_{\mu\nu}\,B^{\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{Q\Delta\mathcal{D}}^{(2)}$ | $(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,\gamma^{\mu}\,\tau^{I}\,Q^{\alpha i}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{I}_{\mu}\,\Delta)]$
$\mathcal{O}_{G\Delta}$ | $G^{A}_{\mu\nu}\,G^{A\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{L\Delta\mathcal{D}}^{(1)}$ | $(N_{f}^{2})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L^{i}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Delta)]$
$\mathcal{O}_{\tilde{G}\Delta}$ | $\tilde{G}^{A}_{\mu\nu}\,G^{A\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{L\Delta\mathcal{D}}^{(2)}$ | $(N_{f}^{2})\,(\overline{L}_{pi}\,\gamma^{\mu}\,\tau^{I}\,L^{i}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}^{I}_{\mu}\,\Delta)]$
$\mathcal{O}_{W\Delta}^{(1)}$ | $W^{I}_{\mu\nu}\,W^{I\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{u\Delta\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u^{\alpha}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Delta)]$
$\mathcal{O}_{W\Delta}^{(2)}$ | $Tr[\Delta^{\dagger}\,W_{\mu\nu}\,\Delta\,W^{\mu\nu}]$ | $\mathcal{O}_{d\Delta\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d^{\alpha}_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Delta)]$
$\mathcal{O}_{\tilde{W}\Delta}^{(1)}$ | $\tilde{W}^{I}_{\mu\nu}\,W^{I\mu\nu}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{e\Delta\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,Tr[(\Delta^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,\Delta)]$
$\mathcal{O}_{\tilde{W}\Delta}^{(2)}$ | $Tr[\Delta^{\dagger}\,W_{\mu\nu}\,\Delta\,\tilde{W}^{\mu\nu}]$ | $\mathcal{O}_{LeH\Delta\mathcal{D}}$ | $(N_{f}^{2})\,Tr[L^{T}_{p}\,C\,i\tau_{2}\,(\gamma^{\mu}\mathcal{D}_{\mu}\,\Delta)\,H\,e_{q}]$
$\mathcal{O}_{BW\Delta}$ | $Tr[\Delta^{\dagger}\,W_{\mu\nu}\,\Delta]\,B^{\mu\nu}$ |  |
$\mathcal{O}_{B\tilde{W}\Delta}$ | $Tr[\Delta^{\dagger}\,\tilde{W}_{\mu\nu}\,\Delta]\,B^{\mu\nu}$ |  |
$\Psi^{2}\Phi^{3}$
$\mathcal{O}_{LeH\Delta}^{(1)}$ | $(N_{f}^{2})\,(\overline{L}_{pi}\,e_{q})\,H^{i}\,Tr[(\Delta^{\dagger}\,\Delta)]$ | $\mathcal{O}_{LeH\Delta}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,(\overline{L}_{pi}\,e_{q})\,\Delta^{\dagger}\,\Delta\,H^{i}}$
$\mathcal{O}_{QdH\Delta}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,d^{\alpha}_{q})\,H^{i}\,Tr[(\Delta^{\dagger}\,\Delta)]}$ | $\mathcal{O}_{QdH\Delta}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,d^{\alpha}_{q})\,\Delta^{\dagger}\,\Delta\,H^{i}}$
$\mathcal{O}_{QuH\Delta}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,\epsilon_{ij}\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q})\,\tilde{H}_{j}\,Tr[(\Delta^{\dagger}\,\Delta)]}$ | $\mathcal{O}_{QuH\Delta}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q})\,\Delta^{\dagger}\,\Delta\,\tilde{H}_{i}}$
$\mathcal{O}_{L\Delta}^{(1)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}+N_{f})\,(L^{T}_{p}\,C\,i\tau_{2}\,\Delta\,L_{q})\,Tr[(\Delta^{\dagger}\,\Delta)]}$ | $\mathcal{O}_{L\Delta}^{(2)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}+N_{f})\,(L^{T}_{p}\,C\,i\tau_{2}\,\Delta\,\Delta^{\dagger}\,\Delta\,L_{q})}$
$\mathcal{O}_{LH\Delta}^{(1)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}+N_{f})\,(L^{T}_{p}\,C\,i\tau_{2}\,\Delta\,L_{q})\,(H^{\dagger}\,H)}$ | $\mathcal{O}_{LH\Delta}^{(2)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,(L^{T}_{pi}\,C\,i\tau_{2}\,\Delta\,H^{i}\,H^{\dagger}_{j}\,L^{j}_{q})}$
$\mathcal{O}_{e\Delta}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}+N_{f})\,(e_{p}^{T}\,C\,e_{q})\,(H^{T}\,\Delta\,H)}$ |  |
$\Psi^{2}\Phi X$
$\mathcal{O}_{WL\Delta}$ | $(N_{f}^{2})\,Tr[(L^{T}_{p}\,C\,i\tau_{2}\,\Delta\,\sigma_{\mu\nu}\,L_{q})\,W^{\mu\nu}]$ | $\mathcal{O}_{BL\Delta}$ | $\boxed{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}(N_{f}^{2}-N_{f})\,(L^{T}_{p}\,C\,i\tau_{2}\,\Delta\,\sigma_{\mu\nu}\,L_{q})\,B^{\mu\nu}}}$
Table 7: SM extended by Complex Triplet Scalar ($\Delta$): Additional
operators of dimension 6. Here $i,j$ and $\alpha$ are the $SU(2)$ and $SU(3)$
indices respectively. $\tau^{I}$ are the $SU(2)$ generators. $A=1,2,\cdots,8$
and $I=1,2,3$. $p,q=1,2,\cdots,N_{f}$ are the flavour indices. Also,
$\Delta=\Delta^{I}\cdot\tau^{I}$ and $W_{\mu\nu}=W^{I}_{\mu\nu}\cdot\tau^{I}$.
Operators in red violate lepton number.

  * •

At dimension 5 in addition to the $\Psi^{2}\Phi^{2}$ class, there are new
operators of the $\Phi^{5}$ class unlike the previous cases.

  * •

Since $\Delta$ transforms as an $SU(2)_{L}$ triplet the
$\Psi^{2}\Phi^{2}\mathcal{D}$ class has operators constituted of higher tensor
products $\mathcal{O}^{(2)}_{L\Delta\mathcal{D}}$ and
$\mathcal{O}^{(2)}_{Q\Delta\mathcal{D}}$ unlike the previous models.

  * •

Lepton number violation too is observed within the
$\Psi^{2}\Phi^{2}\mathcal{D}$, $\Psi^{2}\Phi^{3}$ and $\Psi^{2}\Phi X$
classes. These operators are highlighted in red colour.

#### SM + Left-Handed Triplet Fermion ($\Sigma$)

The extra IR DOF can be fermionic in nature instead of scalar. To demonstrate
the feature of such cases, we have considered a specific example, where the SM
is extended by an $SU(2)_{L}$ real triplet fermion
$\Sigma=(\Sigma_{1},\,\,\Sigma_{2},\,\,\Sigma_{3})$. This additional DOF plays
a central role in the generation of neutrino masses and mixing Foot:1988aq ;
Chakrabortty:2008zh ; Franceschini_2008 ; Cai:2017mow ; Chen:2011de ;
Chakrabortty:2010rq ; Chakrabortty:2010az ; Gu:2011yx , lepton flavour
violating decays Abada:2008ea ; Cheng:1980tp ; Marciano:1977wx ; Eboli:2011ia
; Sirunyan:2017qkz ; ATLAS:2018ghc ; Goswami:2018jar , explaining dark matter
Chaudhuri:2015pna ; Choubey:2017yyn ; Restrepo:2019ilz ; FileviezPerez:2008sr
, and CP & matter-antimatter asymmetry Bajc:2007zf ; Franco:2015pva ;
Branco:2011zb ; Hambye:2012fh . In most of these scenarios, this triplet
fermion is accompanied by other particles which can be integrated out to
construct an effective Lagrangian described by SM DOFs and $\Sigma$. This
motivates us to compute a complete set of effective operators which will
capture all such extended BSM scenarios. Here, we have classified the
effective operators of dimensions 5 and 6 containing $\Sigma$ in Tables 8 and
9 respectively. The operators with distinct hermitian conjugates have been
coloured blue.

Features of the additional operators:

  * •

Since $\Sigma$ has zero hypercharge, in addition to the $\Psi^{2}\Phi^{2}$
operators, we also obtain operators of the class $\Psi^{2}X$ at dimension 5.

  * •

On account of $\Sigma$ being an $SU(2)_{L}$ triplet we obtain multiple
operators of similar structure whenever the other doublets $L,Q,H$ or the
triplet $W^{I}_{\mu\nu}$ are involved.

  * •

We obtain lepton and baryon number violation among operators of the class
$\Psi^{4}$, $\Psi^{2}\Phi X$ and $\Psi^{2}\Phi^{3}$. These operators have been
coloured red.

$\Psi^{2}\Phi^{2}$ | $\Psi^{2}X$
---|---
$\tilde{\mathcal{O}}_{\Sigma H}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{(N_{f}^{2})\,((\Sigma^{I}_{p})^{T}\,C\,\Sigma^{I}_{q})\,(H^{\dagger}\,H)}$ | $\tilde{\mathcal{O}}_{B\Sigma}$ | $\boxed{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\frac{1}{2}(N_{f}^{2}-N_{f})\,B_{\mu\nu}\,((\Sigma^{I}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\Sigma^{I}_{q})}}$
$\tilde{\mathcal{O}}_{e\Sigma H}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{ij}\,(\overline{\Sigma}^{I}_{p}\,e_{q})\,(H^{i}\,\tau^{I}H^{j})}$ | $\tilde{\mathcal{O}}_{W\Sigma}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\frac{1}{2}(N_{f}^{2}+N_{f})\,\epsilon_{IJK}\,W^{I}_{\mu\nu}\,((\Sigma^{J}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\Sigma^{K}_{q})}$
Table 8: SM extended by Left-Handed Triplet Fermion ($\Sigma$): Additional operators of dimension 5. Here $i,j$ are the $SU(2)$ indices. $\tau^{I}$ is the $SU(2)$ generator. $I=1,2,3$ and $p,q=1,2,\cdots,N_{f}$ are the flavour indices. The operator in red violates lepton number. $\Psi^{2}\Phi X$ | $\Psi^{2}\Phi^{2}\mathcal{D}$
---|---
$\mathcal{O}_{BL\Sigma H}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,((L^{i}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\Sigma^{I}_{q})\,\tau^{I}\,H^{j}}$ | $\mathcal{O}_{\Sigma H}^{(1)}$ | $(N_{f}^{2})\,(\overline{\Sigma}^{I}_{p}\gamma^{\mu}\Sigma^{I}_{q})\,(H^{\dagger}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H)$
$\mathcal{O}_{WL\Sigma H}^{(1)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,((L^{i}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\Sigma^{I}_{q})\,H^{j}}$ | $\mathcal{O}_{\Sigma H}^{(2)}$ | $(N_{f}^{2})\,\epsilon_{IJK}\,(\overline{\Sigma}^{I}_{p}\,\gamma^{\mu}\,\Sigma^{J}_{q})\,(H^{\dagger}\,\tau^{K}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H)$
$\mathcal{O}_{WL\Sigma H}^{(2)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{IJK}\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,((L^{i}_{p})^{T}\,C\,\sigma^{\mu\nu}\,\Sigma^{J}_{q})\,\tau^{K}\,H^{j}}$ |  |
$\Psi^{4}$
$\mathcal{O}_{u\Sigma}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma_{\mu}\,u^{\alpha}_{q})\,(\overline{\Sigma}^{I}_{r}\,\gamma^{\mu}\,\Sigma^{I}_{s})$ | $\mathcal{O}_{d\Sigma}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma_{\mu}\,d^{\alpha}_{q})\,(\overline{\Sigma}^{I}_{r}\,\gamma^{\mu}\,\Sigma^{I}_{s})$
$\mathcal{O}_{e\Sigma}$ | $(N_{f}^{4})\,(\overline{e}_{p}\,\gamma_{\mu}\,e_{q})\,(\overline{\Sigma}^{I}_{r}\,\gamma^{\mu}\,\Sigma^{I}_{s})$ | $\mathcal{O}_{\Sigma\Sigma}$ | $(\frac{3}{4}N_{f}^{4}+\frac{1}{2}N_{f}^{3}+\frac{3}{4}N_{f}^{2})\,(\overline{\Sigma}^{I}_{p}\,\gamma_{\mu}\,\Sigma^{I}_{q})\,(\overline{\Sigma}^{J}_{r}\,\gamma^{\mu}\,\Sigma^{J}_{s})$
$\mathcal{O}_{Q\Sigma}^{(1)}$ | $(N_{f}^{4})\,(\overline{Q}_{p\alpha i}\,\gamma_{\mu}\,Q^{\alpha i}_{q})\,(\overline{\Sigma}^{I}_{r}\,\gamma^{\mu}\,\Sigma^{I}_{s})$ | $\mathcal{O}_{Q\Sigma}^{(2)}$ | $(N_{f}^{4})\,\epsilon_{IJK}\,(\overline{Q}_{p\alpha i}\,\gamma_{\mu}\,\tau^{I}\,Q^{j\alpha}_{q})\,(\overline{\Sigma}^{J}_{r}\,\gamma^{\mu}\,\Sigma^{K}_{s})$
$\mathcal{O}_{L\Sigma}^{(1)}$ | $(N_{f}^{4})\,(\overline{L}_{pi}\,\gamma_{\mu}\,L^{i}_{q})\,(\overline{\Sigma}^{I}_{r}\,\gamma^{\mu}\,\Sigma^{I}_{s})$ | $\mathcal{O}_{L\Sigma}^{(2)}$ | $(N_{f}^{4})\,\epsilon_{IJK}\,(\overline{L}_{pi}\,\gamma_{\mu}\,\tau^{I}\,L^{i}_{q})\,(\overline{\Sigma}^{J}_{r}\,\gamma^{\mu}\,\Sigma^{K}_{s})$
$\mathcal{O}_{eL\Sigma}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{4})\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,\tau^{I}\,L^{j}_{q})\,(\overline{e}_{r}\,\Sigma^{I}_{s})}$ | $\mathcal{O}_{\Sigma^{2}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\frac{1}{4}(N_{f}^{4}+3N_{f}^{2})\,((\Sigma^{I}_{p})^{T}\,C\,\Sigma^{I}_{q})\,((\Sigma^{I}_{r})^{T}\,C\,\Sigma^{I}_{s})}$
$\mathcal{O}_{QLd\Sigma}^{(1)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{4})\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,\tau^{I}\,Q^{\alpha j}_{q})\,(\overline{d}_{r\alpha}\,\Sigma^{I}_{s})}$ | $\mathcal{O}_{QLd\Sigma}^{(2)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{4})\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,\sigma_{\mu\nu}\,\tau^{I}\,Q^{\alpha j}_{q})\,(\overline{d}_{r\alpha}\,\sigma^{\mu\nu}\,\Sigma^{I}_{s})}$
$\mathcal{O}_{QLu\Sigma}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{4})\,(\overline{Q}_{p\alpha i}\,u^{\alpha}_{q})\,[\tau^{I}]^{i}_{j}\,((L^{j}_{r})^{T}\,C\,\Sigma^{I}_{s})}$ | $\mathcal{O}_{Qd\Sigma}$ | $\boxed{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}N_{f}^{3}(N_{f}-1)\,\epsilon_{\alpha\beta\gamma}\,\epsilon_{ij}\,(\overline{\Sigma}^{I}_{p}\,d^{\alpha}_{q})\,((Q^{\beta i}_{r})^{T}\,C\,\tau^{I}\,Q^{\gamma j}_{s})}}$
$\Psi^{2}\Phi^{3}$
$\mathcal{O}_{L\Sigma HH}^{(1)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,\Sigma^{I}_{q})\,\tau^{I}\,H^{j}\,(H^{\dagger}\,H)}$ | $\mathcal{O}_{L\Sigma HH}^{(2)}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{(N_{f}^{2})\,\epsilon_{IJK}\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,\Sigma^{I}_{q})\,\tau^{J}\,H^{j}\,(H^{\dagger}\,\tau^{K}\,H)}$
Table 9: SM extended by Left-Handed Triplet Fermion ($\Sigma$): Additional
operators of dimension 6. Here $i,j$ and $\alpha,\beta,\gamma$ are the $SU(2)$
and $SU(3)$ indices respectively. $\tau^{I}$ are the $SU(2)$ generators.
$I,J,K=1,2,3$. $p,q,r,s=1,2,\cdots,N_{f}$ are the flavour indices. Operators
in red violate lepton and baryon numbers.

#### SM + Vector-like Leptons ($V_{L,R},\,E_{L,R},\,N_{L,R}$)

It may be possible that the SM is extended by a set of lighter degrees of
freedom. To discuss that kind of scenario, here, we have considered an example
model where the IR DOFs are vector like leptons: lepton doublets ($V_{L,R}$)
with hypercharge $\frac{1}{2}$, singlets with hypercharge -1 ($E_{L,R}$), and
0 ($N_{L,R}$). This subset of particles can be embedded in a rather complete
scenario where parity is respected, e.g., Pati-Salam, LRSM etc. The vector
like fermions may induce first order Electroweak Phase Transition (EWPT) and
explain the origin of baryon asymmetry Angelescu:2020yzf ; Angelescu_2019 ;
Huo_2015 ; Ishiwata_2015 ; borah2020subtev . They also affect low energy
observables Falkowski_2014 ; Ellis_2014 ; crivellin2020global . The effects of
parity conserving complete theories can be captured through the effective
operators involving these vector like fermions. In that case, it would be
important to note how the tree-level predictions get affected in the presence
of the higher dimensional operators. Here, we have listed the dimension 5
operators in Table 10. We have catalogued the set of dimension 6 operators in
Tables 11-17. The operators with distinct hermitian conjugates are depicted in
blue colour.

Features of the additional operators:

  * •

At dimension 5, we have both $B,L$ conserving and $L$ violating operators
within the $\Psi^{2}\Phi^{2}$ and $\Psi^{2}X$ classes.

  * •

At dimension 6, we have the freedom to write down multiple covariant forms
corresponding to a particular operator. But not all of them are independent.
For example, the operator $\mathcal{O}_{V_{L}V_{R}N_{L}N_{R}}$ in Table 14 can
be written in a covariant form as either
$(\overline{V}_{Rpi}\,V_{Lq}^{i})\,(\overline{N}_{Lr}\,N_{Rq})$ or
$(\overline{V}_{Rpi}\,\gamma^{\mu}\,N_{Rq})\,(\overline{N}_{Lr}\,\gamma_{\mu}\,V_{Ls}^{i})$.
But these two structures are related to each other through the identities
mentioned in subsection 2.3. Therefore, we have included only one of them to
avoid any redundancy in the operator set.

  * •

This model offers the violation of baryon and lepton numbers of different
amount unlike other scenarios discussed in this paper. We have noted the
$(\Delta B,\Delta L)$ of following amounts: $(0,\pm 2)$, $(0,\pm 4)$. $(\pm
1,\pm 1)$ and $(\pm 1,\mp 1)$ within the $\Psi^{4}$, $\Psi^{2}\Phi^{3}$,
$\Psi^{2}\Phi^{2}\mathcal{D}$, and $\Psi^{2}\Phi X$ classes.

$\Psi^{2}\Phi^{2}$
---
$\tilde{\mathcal{O}}_{HE_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,H^{i})\,(\overline{E}_{Rp}\,E_{Lq})$ | $\tilde{\mathcal{O}}_{HN_{L}N_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,H^{i})\,(\overline{N}_{Lp}\,N_{Rq})$
$\tilde{\mathcal{O}}_{HLV_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,H^{i})\,(\overline{L}_{pj}\,V_{Rq}^{j})$ | $\tilde{\mathcal{O}}_{HV_{L}V_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,H^{i})\,(\overline{V}_{Lpj}\,V_{Rq}^{j})$
$\tilde{\mathcal{O}}_{HLV_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,\tau^{I}\,H^{i})\,(\overline{L}_{pj}\,\tau^{I}\,V_{Rq}^{j})$ | $\tilde{\mathcal{O}}_{HV_{L}V_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,\tau^{I}\,H^{i})\,(\overline{V}_{Lpj}\,\tau^{I}\,V_{Rq}^{j})$
$\tilde{\mathcal{O}}_{HeE_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(H^{\dagger}_{i}\,H^{i})\,(\overline{e}_{p}\,E_{Lq})$ | $\tilde{\mathcal{O}}_{HN_{R}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}N_{f}(N_{f}+1)\,(H^{\dagger}_{i}\,H^{i})\,(N_{Rp}^{T}\,C\,N_{Rq})$
$\tilde{\mathcal{O}}_{HV_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}N_{f}(N_{f}+1)\epsilon_{ij}\,\epsilon_{mn}\,H^{i}\,H^{m}\,(V_{Lp}^{j})^{T}C\,V_{Lq}^{n}$ | $\tilde{\mathcal{O}}_{HV_{R}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}N_{f}(N_{f}+1)\epsilon_{ij}\,\epsilon_{mn}\,H^{i}\,H^{m}\,(V_{Rp}^{j})^{T}C\,V_{Rq}^{n}$
$\tilde{\mathcal{O}}_{HLV_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\epsilon_{ij}\,\epsilon_{mn}\,H^{i}\,H^{m}\,(L_{p}^{j})^{T}C\,V_{Lq}^{n}$ | $\tilde{\mathcal{O}}_{HN_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\frac{1}{2}N_{f}(N_{f}+1)\,(H^{\dagger}_{i}\,H^{i})\,(N_{Lp}^{T}\,C\,N_{Lq})$
$\Psi^{2}X$
$\tilde{\mathcal{O}}_{BeE_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B^{\mu\nu}(\overline{e}_{p}\,\sigma_{\mu\nu}\,E_{Lq})$ | $\tilde{\mathcal{O}}_{BE_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B^{\mu\nu}(\overline{E}_{Rp}\,\sigma_{\mu\nu}\,E_{Lq})$
$\tilde{\mathcal{O}}_{BLV_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B^{\mu\nu}(\overline{V}_{Rpi}\,\sigma_{\mu\nu}\,L_{q}^{i})$ | $\tilde{\mathcal{O}}_{WLV_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W_{\mu\nu}^{I}\,(\overline{V}_{Rpi}\,\tau^{I}\sigma^{\mu\nu}\,L_{q}^{i})$
$\tilde{\mathcal{O}}_{BV_{L}V_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B^{\mu\nu}(\overline{V}_{Rpi}\,\sigma_{\mu\nu}\,V_{Lq}^{i})$ | $\tilde{\mathcal{O}}_{WV_{L}V_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W_{\mu\nu}^{I}\,(\overline{V}_{Rpi}\,\tau^{I}\sigma^{\mu\nu}\,V_{Lq}^{i})$
$\tilde{\mathcal{O}}_{BN_{L}N_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B^{\mu\nu}(\overline{N}_{Rp}\,\sigma_{\mu\nu}\,N_{Lq})$ | $\tilde{\mathcal{O}}_{BN_{R}}$ | $\boxed{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}N_{f}(N_{f}-1)\,B_{\mu\nu}\,(N_{Rp}^{T}\,C\,\sigma^{\mu\nu}N_{Rq})}}$
$\tilde{\mathcal{O}}_{BN_{L}}$ | $\boxed{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}{\frac{1}{2}N_{f}(N_{f}-1)\,B_{\mu\nu}\,(N_{Lp}^{T}\,C\,\sigma^{\mu\nu}N_{Lq})}}$ |  |
Table 10: SM extended by Vector-like Leptons ($V_{L,R},\,E_{L,R},\,N_{L,R}$):
Additional operators of dimension 5. $i,j,m,n$ are the $SU(2)$ indices and
$p,q=1,2,\cdots,N_{f}$ are the flavour indices. $\tau^{I}\,(I=1,2,3)$ is the
$SU(2)$ generator. Operators in red violate lepton and baryon numbers.
$\Psi^{2}\Phi^{3}$
---
$\mathcal{O}_{HV_{L}e}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{V}_{Lpi}\,e_{q}\,H^{i})\,(H^{\dagger}_{j}\,H^{j})$ | $\mathcal{O}_{HV_{R}E_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})(\overline{V}_{Rpi}\,E_{Lq}\,H^{i})\,(H^{\dagger}_{j}\,H^{j})$
$\mathcal{O}_{HV_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{V}_{Lpi}\,E_{Rq}\,H^{i})\,(H^{\dagger}_{j}\,H^{j})$ | $\mathcal{O}_{HLE_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{L}_{pi}\,E_{Rq}\,H^{i})\,(H^{\dagger}_{j}\,H^{j})$
$\mathcal{O}_{HV_{L}N_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,(\overline{N}_{Rp}\,V_{Lq}^{i}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$ | $\mathcal{O}_{HLN_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,(\overline{N}_{Rp}\,L_{q}^{i}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$
$\mathcal{O}_{HV_{R}N_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,(\overline{N}_{Lp}\,V_{Rq}^{i}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$ | $\mathcal{O}_{HLN_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,((L^{i}_{p})^{T}\,C\,N_{Lq}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$
$\mathcal{O}_{HV_{L}N_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,((V^{i}_{Lp})^{T}\,C\,N_{Lq}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$ | $\mathcal{O}_{HV_{R}N_{R}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,((V^{i}_{Rp})^{T}\,C\,N_{Rq}\,H^{j})\,(H^{\dagger}_{k}\,H^{k})$
$\Psi^{2}\Phi X$
$\mathcal{O}_{BHV_{L}e}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,e_{q})\,H^{i}$ | $\mathcal{O}_{WHV_{L}e}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W^{I}_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,e_{q})\,\tau^{I}\,H^{i}$
$\mathcal{O}_{BHLE_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,E_{Rq})\,H^{i}$ | $\mathcal{O}_{WHLE_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W^{I}_{\mu\nu}\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,E_{Rq})\,\tau^{I}\,H^{i}$
$\mathcal{O}_{BHLN_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,N_{Rq})\,\tilde{H}_{i}$ | $\mathcal{O}_{WHLN_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W^{I}_{\mu\nu}\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,N_{Rq})\,\tau^{I}\,\tilde{H}_{i}$
$\mathcal{O}_{BHV_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,E_{Rq})\,H^{i}$ | $\mathcal{O}_{WHV_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W^{I}_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,E_{Rq})\,\tau^{I}\,H^{i}$
$\mathcal{O}_{BHV_{L}N_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,(\overline{N}_{Rpi}\,\sigma^{\mu\nu}\,V_{Lq}^{i})\,H^{j}$ | $\mathcal{O}_{WHV_{L}e}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W^{I}_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,e_{q})\,\tau^{I}\,H^{i}$
$\mathcal{O}_{WHV_{L}N_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,W_{\mu\nu}^{I}\,(\overline{N}_{Rpi}\,\sigma^{\mu\nu}\,V_{Lq}^{i})\,\tau^{I}\,H^{j}$ | $\mathcal{O}_{BHV_{L}E_{R}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,E_{Rq})\,H^{i}$
$\mathcal{O}_{WHV_{R}E_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W_{\mu\nu}^{I}\,(\overline{V}_{Rpi}\,\sigma^{\mu\nu}\,E_{Lq})\,\tau^{I}\,H^{i}$ | $\mathcal{O}_{BHV_{R}N_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,B_{\mu\nu}\,(\overline{V}_{Rpi}\,\sigma^{\mu\nu}\,N_{Lq})\,\tilde{H}_{i}$
$\mathcal{O}_{WHV_{R}N_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,W_{\mu\nu}^{I}\,(\overline{V}_{Rpi}\,\sigma^{\mu\nu}\,N_{Lq})\,\tau^{I}\,\tilde{H}_{i}$ | $\mathcal{O}_{BHV_{R}E_{L}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,(\overline{V}_{Rpi}\,\sigma^{\mu\nu}\,E_{Lq})\,H^{i}$
$\mathcal{O}_{BHLN_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,(N_{Lp}^{T}\,C\,\sigma^{\mu\nu}\,L_{q}^{i})\,H^{j}$ | $\mathcal{O}_{BHV_{L}N_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,(N_{Lp}^{T}\,C\,\sigma^{\mu\nu}\,V_{Lq}^{i})\,H^{j}$
$\mathcal{O}_{BHV_{R}N_{R}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,B_{\mu\nu}\,(N_{Rp}^{T}\,C\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,H^{j}$ | $\mathcal{O}_{WHLN_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,(N_{Lp}^{T}\,C\,\sigma^{\mu\nu}\,L_{q}^{i})\,\tau^{I}\,H^{j}$
$\mathcal{O}_{WHV_{R}N_{R}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,(N_{Rp}^{T}\,C\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,\tau^{I}\,H^{j}$ | $\mathcal{O}_{WHV_{L}N_{L}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,W^{I}_{\mu\nu}\,(N_{Lp}^{T}\,C\,\sigma^{\mu\nu}\,V_{Lq}^{i})\,\tau^{I}\,H^{j}$
$\Psi^{2}\Phi^{2}\mathcal{D}$
$\mathcal{O}_{HV_{R}\mathcal{D}}^{(1)}$ | $(N_{f}^{2})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{i})$ | $\mathcal{O}_{HV_{L}\mathcal{D}}^{(1)}$ | $(N_{f}^{2})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,V_{Lq}^{i})\,(H^{\dagger}_{j}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{j})$
$\mathcal{O}_{HV_{R}\mathcal{D}}^{(2)}$ | $(N_{f}^{2})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,\tau^{I}V_{Rq}^{i})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}^{I}\,H^{i})$ | $\mathcal{O}_{HV_{L}i\mathcal{D}}^{(2)}$ | $(N_{f}^{2})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,\tau^{I}V_{Lq}^{i})\,(H^{\dagger}_{j}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{j})$
$\mathcal{O}_{HN_{L}\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{N}_{Lp}\,\gamma^{\mu}\,N_{Lq})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{i})$ | $\mathcal{O}_{HN_{R}\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{N}_{Rp}\,\gamma^{\mu}\,N_{Rq})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{i})$
$\mathcal{O}_{HE_{L}\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{i})$ | $\mathcal{O}_{HE_{R}\mathcal{D}}$ | $(N_{f}^{2})\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(H^{\dagger}_{i}\,i\overleftrightarrow{\mathcal{D}}_{\mu}\,H^{i})$
$\mathcal{O}_{HeE_{R}\mathcal{D}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{e}_{p}\,\gamma^{\mu}\,E_{Rq})\,(H^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$ | $\mathcal{O}_{HeN_{R}\mathcal{D}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{N}_{Rp}\,\gamma^{\mu}\,e_{q})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$
$\mathcal{O}_{HE_{L}N_{L}\mathcal{D}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,\epsilon_{ij}\,(\overline{N}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$ | $\mathcal{O}_{HLV_{L}\mathcal{D}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{L}_{pi}\,\gamma^{\mu}\,V_{Lq}^{i})\,(H^{\dagger}_{j}\,i\mathcal{D}_{\mu}\,H^{j})$
$\mathcal{O}_{HE_{R}N_{R}\mathcal{D}}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{N}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$ | $\mathcal{O}_{HLV_{L}\mathcal{D}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{2})\,(\overline{L}_{pi}\,\gamma^{\mu}\,\tau^{I}V_{Lq}^{i})\,(H^{\dagger}_{j}\,i\mathcal{D}_{\mu}^{I}\,H^{j})$
$\mathcal{O}_{HLV_{R}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,\epsilon_{mn}\,((V^{i}_{Rp})^{T}\,C\,\gamma^{\mu}\,L^{m}_{q})\,(\tilde{H}^{\dagger}_{n}\,i\mathcal{D}_{\mu}\,H^{j})$ | $\mathcal{O}_{HV_{L}V_{R}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,\epsilon_{ij}\,\epsilon_{mn}\,((V^{i}_{Rp})^{T}\,C\,\gamma^{\mu}\,V^{m}_{Lq})\,(\tilde{H}^{\dagger}_{n}\,i\mathcal{D}_{\mu}\,H^{j})$
$\mathcal{O}_{HeN_{L}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,(e^{T}_{p}\,C\,\gamma^{\mu}\,N_{Lq})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$ | $\mathcal{O}_{HE_{R}N_{L}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,(E^{T}_{Rp}\,C\,\gamma^{\mu}\,N_{Lq})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$
$\mathcal{O}_{HE_{L}N_{R}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,(E^{T}_{Lp}\,C\,\gamma^{\mu}\,N_{Rq})\,(\tilde{H}^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$ | $\mathcal{O}_{HN_{L}N_{R}\mathcal{D}}$ | $\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}(N_{f}^{2})\,(N^{T}_{Lp}\,C\,\gamma^{\mu}\,N_{Rq})\,(H^{\dagger}_{i}\,i\mathcal{D}_{\mu}\,H^{i})$
Table 11: SM extended by Vector-like Leptons ($V_{L,R},\,E_{L,R},\,N_{L,R}$):
Additional operators of dimension 6. $i,j,m,n$ are the $SU(2)$ indices and
$p,q=1,2,\cdots,N_{f}$ are the flavour indices. $\tau^{I}\,(I=1,2,3)$ is the
$SU(2)$ generator. Operators in red violate lepton number. $\Psi^{4}$
---
$\mathcal{O}_{dE_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{d}_{r\alpha}\,\gamma_{\mu}\,d_{s}^{\alpha})$ | $\mathcal{O}_{eE_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{e}_{r}\,\gamma_{\mu}\,e_{s})$
$\mathcal{O}_{eE_{R}}$ | $(N_{f}^{4})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$ | $\mathcal{O}_{\bar{E}_{L}\bar{E}_{R}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$
$\mathcal{O}_{E_{R}}$ | $\frac{1}{4}N_{f}^{2}(N_{f}+1)^{2}\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$ | $\mathcal{O}_{E_{L}}$ | $\frac{1}{4}N_{f}^{2}(N_{f}+1)^{2}\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})(\overline{E}_{Lr}\,\gamma_{\mu}\,E_{Ls})$
$\mathcal{O}_{LE_{L}}$ | $(N_{f}^{4})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L_{q}^{i})(\overline{E}_{Lr}\,\gamma_{\mu}\,E_{Ls})$ | $\mathcal{O}_{LE_{R}}$ | $(N_{f}^{4})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L_{q}^{i})(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$
$\mathcal{O}_{LV_{L}}^{(1)}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,V_{Lq}^{i})\,(\overline{L}_{rj}\,\gamma_{\mu}L_{s}^{j})$ | $\mathcal{O}_{V_{L}E_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{V}_{Lri}\,\gamma_{\mu}\,V_{Ls}^{i})$
$\mathcal{O}_{LV_{L}}^{(2)}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,\tau^{I}\,V_{Lq}^{i})\,(\overline{L}_{rj}\,\gamma_{\mu}\,\tau^{I}\,L_{s}^{j})$ | $\mathcal{O}_{V_{L}E_{R}}$ | $(N_{f}^{4})\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(\overline{V}_{Lri}\,\gamma_{\mu}\,V_{Ls}^{i})$
$\mathcal{O}_{dV_{L}}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d_{q}^{\alpha}\,)(\overline{V}_{Lri}\,\gamma_{\mu}\,V_{Ls}^{i})$ | $\mathcal{O}_{V_{L}e}$ | $(N_{f}^{4})(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\overline{V}_{Lri}\,\gamma_{\mu}\,V_{Ls}^{i})$
$\mathcal{O}_{V_{L}}$ | $\frac{1}{2}N_{f}^{2}(N_{f}^{2}+1)\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,V_{Lq}^{i}\,)(\overline{V}_{Lrj}\,\gamma_{\mu}\,V_{Ls}^{j})$ | $\mathcal{O}_{dV_{R}}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d_{q}^{\alpha}\,)(\overline{V}_{Rri}\,\gamma_{\mu}\,V_{Rs}^{i})$
$\mathcal{O}_{V_{R}E_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{V}_{Rri}\,\gamma_{\mu}\,V_{Rs}^{i})$ | $\mathcal{O}_{eV_{R}}$ | $(N_{f}^{4})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\overline{V}_{Rri}\,\gamma_{\mu}\,V_{Rs}^{i})$
$\mathcal{O}_{V_{R}E_{R}}$ | $(N_{f}^{4})\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(\overline{V}_{Rri}\,\gamma_{\mu}\,V_{Rs}^{i})$ | $\mathcal{O}_{V_{R}}$ | $\frac{1}{2}N_{f}^{2}(N_{f}^{2}+1)\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i}\,)(\overline{V}_{Rrj}\,\gamma_{\mu}\,V_{Rs}^{j})$
$\mathcal{O}_{LV_{R}}^{(1)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i})\,(\overline{L}_{rj}\,\gamma_{\mu}\,L_{s}^{j})$ | $\mathcal{O}_{V_{L}V_{R}}^{(1)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i})\,(\overline{V}_{Lrj}\,\gamma_{\mu}\,V_{Ls}^{j})$
$\mathcal{O}_{LV_{R}}^{(2)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,\tau^{I}\,V_{Rq}^{i})\,(\overline{L}_{rj}\,\gamma_{\mu}\,\tau^{I}\,L_{s}^{j})$ | $\mathcal{O}_{V_{L}V_{R}}^{(2)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,\tau^{I}\,V_{Rq}^{i})\,(\overline{V}_{Lrj}\,\gamma^{\mu}\,\tau^{I}\,V_{Ls}^{j})$
$\mathcal{O}_{dN_{L}}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d_{q}^{\alpha}\,)(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{eN_{L}}$ | $(N_{f}^{4})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$
$\mathcal{O}_{E_{L}N_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{E_{R}N_{L}}$ | $(N_{f}^{4})\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$
$\mathcal{O}_{LN_{L}}$ | $(N_{f}^{4})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L_{q}^{i})(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{V_{L}N_{L}}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,V_{Lq}^{i})(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$
$\mathcal{O}_{V_{R}N_{L}}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i})(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{N_{L}}$ | $\frac{1}{4}N_{f}^{2}(N_{f}+1)^{2}\,(\overline{N}_{Lp}\,\gamma^{\mu}\,N_{Lq})(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$
$\mathcal{O}_{dN_{R}}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d_{q}^{\alpha}\,)(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{eN_{R}}$ | $(N_{f}^{4})\,(\overline{e}_{p}\,\gamma^{\mu}\,e_{q})\,(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$
$\mathcal{O}_{E_{L}N_{R}}$ | $(N_{f}^{4})\,(\overline{E}_{Lp}\,\gamma^{\mu}\,E_{Lq})\,(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{E_{R}N_{R}}$ | $(N_{f}^{4})\,(\overline{E}_{Rp}\,\gamma^{\mu}\,E_{Rq})\,(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$
$\mathcal{O}_{LN_{R}}$ | $(N_{f}^{4})\,(\overline{L}_{pi}\,\gamma^{\mu}\,L_{q}^{i})(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{V_{L}N_{R}}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma^{\mu}\,V_{Lq}^{i})(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$
$\mathcal{O}_{V_{R}N_{R}}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma^{\mu}\,V_{Rq}^{i})(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{N_{L}N_{R}}$ | $(N_{f}^{4})\,(\overline{N}_{Lp}\,\gamma^{\mu}\,N_{Lq})\,(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$
$\mathcal{O}_{N_{R}}$ | $\frac{1}{4}N_{f}^{2}(N_{f}+1)^{2}(\overline{N}_{Rp}\,\gamma^{\mu}\,N_{Rq})(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{QE_{L}}$ | $(N_{f}^{4})\,(\overline{Q}_{\alpha p}\,\gamma^{\mu}\,Q_{q}^{\alpha})(\overline{E}_{Lr}\,\gamma_{\mu}\,E_{Ls})$
$\mathcal{O}_{QN_{L}}$ | $(N_{f}^{4})\,(\overline{Q}_{\alpha p}\,\gamma^{\mu}\,Q_{q}^{\alpha})(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{QE_{R}}$ | $(N_{f}^{4})\,(\overline{Q}_{\alpha p}\,\gamma^{\mu}\,Q_{q}^{\alpha})(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$
$\mathcal{O}_{QV_{L}}^{(1)}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma_{\mu}\,V_{Lq}^{i})\,(\overline{Q}_{r\alpha j}\,\gamma^{\mu}\,Q_{s}^{\alpha j})$ | $\mathcal{O}_{QV_{R}}^{(1)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma_{\mu}\,V_{Rq}^{i})\,(\overline{Q}_{rj\alpha}\,\gamma^{\mu}\,Q_{s}^{j\alpha})$
$\mathcal{O}_{QV_{L}}^{(2)}$ | $(N_{f}^{4})\,(\overline{V}_{Lpi}\,\gamma_{\mu}\,\tau^{I}\,V_{Lq}^{i})\,(\overline{Q}_{rj\alpha}\,\gamma^{\mu}\,\tau^{I}\,Q_{s}^{j\alpha})$ | $\mathcal{O}_{QV_{R}}^{(2)}$ | $(N_{f}^{4})\,(\overline{V}_{Rpi}\,\gamma_{\mu}\,\tau^{I}\,V_{Rq}^{i})\,(\overline{Q}_{rj\alpha}\,\gamma^{\mu}\,\tau^{I}\,Q_{s}^{j\alpha})$
$\mathcal{O}_{QN_{R}}$ | $(N_{f}^{4})\,(\overline{Q}_{p\alpha}\,\gamma^{\mu}\,Q_{q}^{\alpha})(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{uE_{L}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{E}_{Lr}\,\gamma_{\mu}\,E_{Ls})$
$\mathcal{O}_{uV_{L}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{V}_{Lr}\,\gamma_{\mu}\,V_{Ls})$ | $\mathcal{O}_{uE_{R}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$
$\mathcal{O}_{uV_{R}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{V}_{Rri}\,\gamma_{\mu}\,V_{Rs}^{i})$ | $\mathcal{O}_{uN_{R}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{N}_{Rr}\,\gamma_{\mu}\,N_{Rs})$
$\mathcal{O}_{uN_{L}}$ | $(N_{f}^{4})\,(\overline{u}_{p\alpha}\,\gamma^{\mu}\,u_{q}^{\alpha})\,(\overline{N}_{Lr}\,\gamma_{\mu}\,N_{Ls})$ | $\mathcal{O}_{dE_{R}}$ | $(N_{f}^{4})\,(\overline{d}_{p\alpha}\,\gamma^{\mu}\,d_{q}^{\alpha}\,)(\overline{E}_{Rr}\,\gamma_{\mu}\,E_{Rs})$
Table 12: SM extended by Vector-like Leptons ($V_{L,R},\,E_{L,R},\,N_{L,R}$):
Additional operators of dimension 6. $i,j$ and $\alpha$ are the $SU(2)$ and
$SU(3)$ indices respectively. $p,q,r,s=1,2,\cdots,N_{f}$ are the flavour
indices. $\tau^{I}\,(I=1,2,3)$ is the $SU(2)$ generator. All the operators in
this table conserve the lepton and baryon numbers $(\Delta B=0,\Delta L=0)$.
$\Psi^{4}$
---
$\mathcal{O}_{V_{L}V_{R}eE_{L}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,e_{s})$ | $\mathcal{O}_{V_{L}V_{R}E_{L}E_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,E_{Rs})$
$\mathcal{O}_{V_{L}V_{R}eE_{L}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,e_{s})$ | $\mathcal{O}_{V_{L}V_{R}E_{L}E_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,E_{Rs})$
$\mathcal{O}_{LV_{L}eN_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{V}_{Lpi}\,e_{q})\,(\overline{L}_{rj}\,\gamma_{\mu}\,N_{Rs})$ | $\mathcal{O}_{LV_{L}E_{R}N_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{V}_{Lpi}\,E_{Rq})\,(\overline{L}_{rj}\,N_{Rs})$
$\mathcal{O}_{LV_{L}eN_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,e_{q})\,(\overline{L}_{rj}\,\sigma_{\mu\nu}\,N_{Rs})$ | $\mathcal{O}_{LV_{L}E_{R}N_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,E_{Rq})\,(\overline{L}_{rj}\,\sigma_{\mu\nu}\,N_{Rs})$
$\mathcal{O}_{LV_{R}eE_{L}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,e_{s})$ | $\mathcal{O}_{LV_{R}E_{L}E_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,E_{Rs})$
$\mathcal{O}_{LV_{R}eE_{L}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,e_{s})$ | $\mathcal{O}_{LV_{R}E_{L}E_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,E_{Rs})$
$\mathcal{O}_{\bar{L}\bar{V}_{L}V_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,V_{Rq}^{i})\,(\overline{L}_{rj}\,V_{Rs}^{j})$ | $\mathcal{O}_{QdV_{R}E_{L}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{d}_{\alpha p}\,Q_{\alpha q}^{i})\,(\overline{V}_{Rri}\,E_{Ls})$
$\mathcal{O}_{\bar{L}\bar{V}_{L}V_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{L}_{rj}\,\sigma_{\mu\nu}\,V_{Rs}^{j})$ | $\mathcal{O}_{QdV_{R}E_{L}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{d}_{\alpha p}\,\sigma^{\mu\nu}\,Q_{\alpha q}^{i})\,(\overline{V}_{Rri}\,\sigma_{\mu\nu}\,E_{Ls})$
$\mathcal{O}_{LV_{R}N_{L}N_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,V_{Rq}^{i})\,(\overline{N}_{Lr}\,N_{Rs})$ | $\mathcal{O}_{\overline{L}V_{L}\overline{V}_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Rpi}\,V_{Lq}^{i})\,(\overline{L}_{rj}\,V_{Rs}^{j})$
$\mathcal{O}_{LV_{R}N_{L}N_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{L}_{pi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{N}_{Lr}\,\sigma_{\mu\nu}\,N_{Rs})$ | $\mathcal{O}_{\overline{L}V_{L}\overline{V}_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Rpi}\,\sigma^{\mu\nu}\,V_{Lq}^{i})\,(\overline{L}_{rj}\,\sigma_{\mu\nu}\,V_{Rs}^{j})$
$\mathcal{O}_{eE_{L}N_{L}N_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{N}_{Lp}\,N_{Rq})\,(\overline{E}_{Lr}\,e_{s})$ | $\mathcal{O}_{N_{L}V_{R}E_{L}E_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{N}_{Lp}\,N_{Rq})\,(\overline{E}_{Lr}\,E_{Rs})$
$\mathcal{O}_{eE_{L}N_{L}N_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{N}_{Lp}\,\sigma^{\mu\nu}\,N_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,e_{s})$ | $\mathcal{O}_{N_{L}N_{R}E_{L}E_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{N}_{Lpi}\,\sigma^{\mu\nu}\,N_{Rq}^{i})\,(\overline{E}_{Lr}\,\sigma_{\mu\nu}\,E_{Rs})$
$\mathcal{O}_{V_{L}V_{R}N_{L}N_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,V_{Rq}^{i})\,(\overline{N}_{Lr}\,N_{Rs})$ | $\mathcal{O}_{QdLN_{R}}^{(1)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{Q}_{\alpha pi}\,d_{\alpha q})\,(\overline{L}_{rj}\,N_{Rs})$
$\mathcal{O}_{V_{L}V_{R}N_{L}N_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,(\overline{V}_{Lpi}\,\sigma^{\mu\nu}\,V_{Rq}^{i})\,(\overline{N}_{Lr}\,\sigma_{\mu\nu}\,N_{Rs})$ | $\mathcal{O}_{QdLN_{R}}^{(2)}$ | $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(N_{f}^{4})\,\epsilon_{ij}\,(\overline{Q}_{\alpha pi}\,\sigma^{\mu\nu}\,d_{\alpha q})\,(\overline{L}_{rj}\,\sigma_{\mu\nu}\,N_{Rs})$