Renormalisation of SMEFT bosonic interactions
up to dimension eight by LNV operators

Supratim Das<EMAIL_ADDRESS>and Álvaro Díaz-
<EMAIL_ADDRESS>
a CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de
Granada, Campus de Fuentenueva, E–18071 Granada, Spain

###### Abstract

We present the renormalisation group running of the bosonic operators of the
Standard Model effective field theory by the Lepton Number Violating operators
(LNVs) at 1-loop order up to $\mathcal{O}(v^{4}/\Lambda^{4})$, with $v\sim
246$ GeV as the electroweak scale and $\Lambda$ as the SMEFT cut-off. Using
these relations with the positivity bounds on Wilson coefficients of
$\phi^{4}D^{4}$ class, we derive sign-constraints on the Wilson coefficients
of LNV operators. We inspect these constraints for the LNV Wilson coefficients
generated from matching the Type-I and III seesaw models to SMEFT up to
dimension seven at tree-level. We also exhibit the unique bounds induced by
the T-parameter on LNVs.

###### Contents

  1. 1 Introduction
  2. 2 Theory and methodology
    1. 2.1 SMEFT conventions
    2. 2.2 Organizing the calculation
      1. 2.2.1 Insertions of $d_{5}\times d_{7}$
      2. 2.2.2 Insertions of $d_{5}\times d_{5}\times d_{6}$
      3. 2.2.3 Insertions of $d_{5}\times d_{5}\times d_{5}\times d_{5}$
      4. 2.2.4 Renormalisation below dimension-8
  3. 3 The RGEs
    1. 3.1 Divergences
    2. 3.2 Explicit expression for the RGEs
    3. 3.3 Detailed example
  4. 4 Discussions and outlook
    1. 4.1 Positivity bounds
    2. 4.2 Constraints from T-parameter
  5. 5 Conclusions
  6. A Tables of operators

##  1 Introduction

The Standard Model Effective Field Theory (SMEFT) is a robust framework to
parametrise New Physics (NP), as it extends the Standard Model (SM) with
effective interactions accounting for the experimental observations supporting
physics beyond SM (BSM) [1]. These effective interactions formed out of the SM
particle content respect the gauge symmetry $SU(3)_{C}\otimes SU(2)_{L}\otimes
U(1)_{Y}$ and are suppressed by the SMEFT cut-off scale. In presence of these
effective operators, the modifications to the observables are parameterised in
terms of the SMEFT Wilson Coefficients (WCs) which encapsulate the deviations
from the SM [2, 3, 4, 5, 6, 7, 8, 9].

To analyze NP interactions via the observables defined at separated energy
scales, one needs to relate WC from these scales consistently. In the present
scenario, where the NP is apparently beyond the reach of current searches, it
is sensible to assume that the NP scale is much larger than that of the
Electroweak (EW) Theory. In a generic (top-down) EFT scenario where NP models
are matched to SMEFT, the WCs of effective operators defined at high-scale
encapsulate the NP interactions and scale. The WCs encode the couplings
featured by a theory defined at a certain energy level. Renormalisation Group
Equations (RGEs) are key to expressing these WCs at energies accessible to the
measurements. We construct these relations for SMEFT, and for this we consider
a finite experimental resolution, which facilitates a truncation on the
infinite tower of effective operators suppressed by according powers of cut-
off ($\Lambda$).

Recent studies indicate that the SMEFT truncation at order $1/\Lambda^{2}$ is
not sufficient for making consistent prediction in several cases. For
instance, dimension-8 interactions are leading contributors in many
observables [10, 11, 5, 12, 13]. In the weakly-coupled Ultraviolet (UV)
completions of the SMEFT, certain dimension-6 classes of operators are not
generated at tree-level [14, 15, 16, 17]. The contribution from dimension-8
(and dimension-6 squared) effects are significant when the SMEFT cut-off is
chosen relatively low [18, 19]. Order $1/\Lambda^{4}$ is also needed in
certain cases where the observables are measured so precisely that they become
sensitive to higher mass dimension operators’ effects [20, 21, 7].

These studies motivate to extend the SMEFT analysis to order $1/\Lambda^{4}$
by including dimension-8 operators effects, which is the method followed in
many recent theoretical studies [22, 23, 13, 24, 25, 7]. Renormalisation up to
order $1/\Lambda^{4}$ in the SMEFT from two dimension-6 operators [27, 29],
and one dimension-8 operators [26, 28] have been studied in detail. Our aim in
this paper is to compute the renormalisation of SMEFT bosonic interactions up
to dimension-8 by including the running induced by lepton number violating
operators up to dimension-7. With LNV operators included in the RGEs, all the
possible contributions to bosonic operators of the SMEFT are known explicitly
up to order $1/\Lambda^{4}$ at 1-loop. There are several reasons to approach
this problem. For instance, the running of some operators can provide the
leading SMEFT corrections to SM predictions to observables in which loop-
induced interactions appear at leading order. Also, under certain conditions,
constraints on dimension-8 operators can reveal unique bounds on the space of
LNV WCs. We discuss one such case where the T-parameter bound is translated to
LNV operators.

Dimension-8 Wilson coefficients are constrained by positivity bounds which are
restrictions on S-matrix elements deduced from the fundamental axioms –
analyticity, unitarity and crossing symmetry on the underlying theory. [30,
31, 32, 33, 34, 35]. Recently, the RGEs of effective operators and its
implications on positivity bounds have been analyzed [36, 28, 37]. Following
the top-down approach, SMEFT WCs are related to model dependent NP couplings.
The RGE effects and the positivity bounds on these WCs impose significant
restrictions on a case-to-case basis on the new physics parameter space. In
this work, we show unique constraints on the UV space deduced solely from the
RGE computation in the IR (SMEFT in this case) and positivity bounds. We also
examine these constraints on two BSMs – Type-I and III seesaw models by first
matching these to SMEFT up to dimension-7 at tree-level, and then, validating
the consistency of derived constraints.

This article is arranged as follows. In Section 2, we establish the relevant
Lagrangian, set the conventions used in the article, and discuss the approach
we follow for computing the renormalisation of the operators involved. In
Section 3, we discuss the derivation of the divergences and RGEs and provide
these at 1-loop up to order $1/\Lambda^{4}$. We point out the global structure
of the RGEs, and comment on contributions that are larger than expected from
naive dimensional analysis. In Section 4, we derive generic constraints on LNV
parameter space using RGEs and positivity bounds, and perform a case study on
the seesaw models. We also discuss bounds induced from the T-parameter to
LNVs. Finally, in Section 5, we summarize our findings. We tabulate all the
effective operators relevant for our analysis in the Appendix A.

##  2 Theory and methodology

###  2.1 SMEFT conventions

We consider the SMEFT Lagrangian,

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\left(\alpha_{\ell\phi}\frac{O_{\ell\phi}^{(5)}}{\Lambda}+\text{h.c.}\right)+\sum_{i}\beta_{i}\frac{O_{i}^{(6)}}{\Lambda^{2}}+\sum_{j}\omega_{j}\frac{O_{j}^{(7)}}{\Lambda^{3}}+\sum_{k}c_{k}\frac{O_{k}^{(8)}}{\Lambda^{4}}\,,$
(2.1)

where the expansion in the SMEFT cut-off scale $\Lambda$ is truncated at order
$1/\Lambda^{4}$ and $\alpha_{\ell\phi}$, $\beta_{i}$, $\omega_{j}$, $c_{k}$
represent $d_{5},\,d_{6},\,d_{7},$ and $d_{8}$ Wilson coefficients,
respectively. The Standard Model Lagrangian is written as:

$\displaystyle\mathcal{L}_{\text{SM}}=$
$\displaystyle-\frac{1}{4}G_{\mu\nu}^{A}G^{A\,\mu\nu}-\frac{1}{4}W_{\mu\nu}^{I}W^{I\,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
$\displaystyle+\overline{q^{m}}\mathrm{i}\not{D}q^{m}+\overline{\ell^{m}}\mathrm{i}\not{D}\ell^{m}+\overline{u^{m}}\mathrm{i}\not{D}u^{m}+\overline{d^{m}}\mathrm{i}\not{D}d^{m}+\overline{e^{m}}\mathrm{i}\not{D}e^{m}$
$\displaystyle+\left(D_{\mu}\phi\right)^{\dagger}\left(D^{\mu}\phi\right)+\mu_{\phi}^{2}|\phi|^{2}-\lambda_{\phi}|\phi|^{4}-\left(y^{u}_{mn}\overline{q^{m}}\widetilde{\phi}u^{n}+y^{d}_{mn}\overline{q^{m}}\phi
d^{n}+y^{e}_{mn}\overline{\ell^{m}}\phi e^{n}+\text{h.c.}\right)\,.$ (2.2)

The covariant derivative is defined as:

$D_{\mu}=\partial_{\mu}-\mathrm{i}g_{1}YB_{\mu}-\mathrm{i}g_{2}\frac{\sigma^{I}}{2}W_{\mu}^{I}-\mathrm{i}g_{3}\frac{\lambda^{A}}{2}G_{\mu}^{A}\,.$
(2.3)

Here, $e$, $u$ and $d$ represent the right-handed leptons, up- and down-type
quarks; $\ell$ and $q$ represent the left-handed leptons and quarks; and
$B,W,$ and $G$ represent the gauge fields corresponding to $U(1)_{Y},\
SU(2)_{L}$ and $SU(3)_{C}$, with $g_{1},\,g_{2}$ and $g_{3}$ as their gauge
couplings, respectively. We represent the SM Higgs doublet by $\phi$
$(Y=\frac{1}{2})$, and $\widetilde{\phi}=\mathrm{i}\sigma^{2}\phi^{\ast}$,
where $\sigma$’s denote the Pauli Matrices $(I=1,2,3)$. We denote the
$U(1)_{Y}$ hypercharge by $Y$, and $\lambda$’s are the Gell-Mann matrices
($A=1,...,8$).

In the SMEFT expansion, the dimension-8 Wilson coefficients are suppressed by
a power of $1/\Lambda^{4}$. From power counting, we deduce that the running of
dimension-8 at this order is triggered by the insertions of (I) one
dimension-8 ($d_{8}$) [28], (II) two dimension-6 ($d_{6}$) [27], (III) one
dimension-5 ($d_{5}$) and one dimension-7 ($d_{7}$), (IV) two dimension-5 and
one dimension-6 or (V) four dimension-5 Wilson coefficients. The Weinberg
operator is the only operator at $d_{5}$ [2]. We use physical bases for the
insertions of $d_{6}$ [38] and $d_{7}$ [39]. However, the method employed to
compute the divergences requires Green’s bases for $d_{6}$ [40] and $d_{8}$
[41] 333Other choices for SMEFT $d_{7}$ and $d_{8}$ operators basis are
available at [42, 43]..

The renormalisation of the dimension-8 Wilson coefficients is performed using
an off-shell scheme [41]. The rationale of this method consists in considering
only diagrams that are 1-particle irreducible (1PI) to generate the
divergences of the Wilson coefficients (including those of non-physical
operators). In general, we could do this by expanding the physical basis with
any set of redundant operators spanning all the possible interactions.

The 1PI diagrams are computed off-shell to generate amplitudes, which are then
captured in the Wilson coefficients of the Green’s basis operators. The
redundant operators can be reduced to the physical subset by applying field
redefinitions or the Equations of Motion (EoMs). Consequently, one finds that
the contribution of the redundant operators amounts to a shift in the
definition of the Wilson coefficients of the physical operators:

$\frac{c_{i}^{\text{phys}}}{\Lambda^{n}}\rightarrow\frac{c_{i}^{\text{phys}}}{\Lambda^{n}}+\frac{1}{\Lambda^{n}}\sum_{k}b_{k}c_{k}^{\text{red}}\,,$
(2.4)

where $b_{k}$ usually contains powers of the SM couplings but can also enclose
SMEFT coefficients of lower dimension provided the mass dimension of each term
in the shift is the same. Combinations of more than one SMEFT WC appear when
the EoMs for some field are applied at a greater order in $\Lambda$. For
$d_{8}$ WCs, this means that the shift of the physical coefficients contain
linear combinations of redundant $d_{8}$ WCs but also pairs of redundant
$d_{6}$ WCs. In principle we could also find contributions of $d_{5}$ and
$d_{7}$ WCs but applying the EoMs to these operators leads only to fermionic
operators, which are out of the scope of this paper. Hence, the only redundant
degrees of freedom we need to consider are those contained in the Green’s
bases for $d_{6}$ and $d_{8}$.

The computation is feasible without introducing the redundant operators.
However, it would imply dealing with all connected diagrams, whereas the off-
shell approach just adds an extra systematical step translating the WCs from
the Green’s basis to the physical basis, and it is already worked out for
bosonic interactions in [41]. In practice, the contribution from connected
diagrams in the on-shell approach is indirectly taken care of when the WC
shift is applied, since it contains the information given by the EoMs.

###  2.2 Organizing the calculation

The main goal of this paper is to compute the contributions of all LNV
operators to the RGEs of the dimension-8 bosonic Wilson coefficients and to
order $1/\Lambda^{4}$. At first, one may think of the many possible different
topologies generated by combinations of $d_{5}$, $d_{6}$ and $d_{7}$ operators
and SM couplings. Considering the great number of operators (1+20+63) can lead
to several valid diagrams, it seems inevitable that the amplitudes for some
processes get contributions from a number of operators. Restricting to bosonic
processes remove a significant number of these and -after a deeper analysis-
we show there are a number of these valid contributions that actually vanish,
thus remarkably reducing the effort needed for the whole computation. In table
1 we schematically show the variety of possibilities and the diagrams we are
left with. The logic on which we discard most of these contributions is
discussed throughout the rest of this section.

$1/\Lambda^{4}$ | $d_{5}^{4}$ | $d_{5}^{2}\times d_{6}$ | $d_{5}\times d_{7}$
---|---|---|---
8-Higgs | 1(a) | - | -
6-Higgs | - | 1(b) |  1(d), 1(i), 1(j)
4-Higgs | - | - |  1(c), 1(e), 1(f), 1(g), 1(h)
2-Higgs | - | - | -
0-Higgs | - | - | -
Table 1:  Relevant contributions for the renormalisation of bosonic operators
to order $\mathcal{O}(v^{4}/\Lambda^{4})$. Rows represent the classes of
dimension-8 bosonic operators, grouped by the number of Higgs they contain.
Columns represent the possibilities for insertion of LNV operators. A number
means the contribution is sizeable and considered in this paper in the Figure
corresponding to that number. A hyphen (-) means the contribution vanishes or
is out of the scope of this paper. See the text for more details.

As we can see in the Lagrangian Eq. 2.1, the WCs are suppressed by a power of
the energy scale $\Lambda$ related to the mass dimension of the operator it
refers to. With this Lagrangian and to order $1/\Lambda^{4}$ the following
contributions need to be considered: $(d_{5}^{4})$, $(d_{5}^{2}\times d_{6})$
and $(d_{5}\times d_{7})$. The divergences of the bosonic $d_{8}$ operators
with $d_{6}^{2}$ and the self-contribution of $d_{8}$ are available in [27]
and [28], respectively, so we do not consider them anymore. What remains is
precisely the aim of this work: contributions including at least one LNV
operator. Before discussing each set of insertions individually, there are
some global remarks that apply to all cases.

A priori, we deal with insertions of $d_{5}$, $d_{6}$ and $d_{7}$ combined as
$d_{5}^{4}$, $d_{5}^{2}\times d_{6}$ and $d_{5}\times d_{7}$ to make 1PI
diagrams of processes that only include bosons as external fields, as we
renormalise the bosonic dimension-8 operators. That would be the maximum
number of contributions we would have to compute, but now we argue that most
of the calculations vanish, and the nonzero combinations are but a small
subset of the total amount of possibilities.

First of all, notice the field content of the loops in the diagrams can only
consist of fermions, and not bosons. It follows from that all of the LNV
operators inserted obviously include one or two pairs of fermion fields (a
bispinor and its conjugate for each pair), which generate a vertex with two or
four fermionic legs. Since the operators we are striving to renormalise do not
contain fermions, then the fermionic legs cannot be external, thus forcing
them to be closed in a loop. This directly affects the renormalisation
operators of mass dimension lower than eight (See subsection 2.2.4).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 1:  Contributions from all possible insertions of LNV operators to
order $1/\Lambda^{4}$. Vertices in red, green and blue represent insertions of
a dimension-5, dimension-6, and dimension-7 operators, respectively. Dashed,
solid, and wavy lines represent scalar, fermion, and gauge boson
propagators/legs respectively.

(a)

(b)

(c)

(d)

(e)

Figure 2:  Relevant diagrams with insertions of LNV operators excluded for two
reasons. Diagrams 2(a) and 2(b) vanish due to symmetries. 2(c) is included in
the Wilson coefficient shift. 2(d) and 2(e) are also included in the Wilson
coefficient shift, but their overall contribution vanishes after computing all
the divergences. Vertices in red, green and blue represent insertions of a
dimension-5, dimension-6, or dimension-7 operators, respectively.

The point made above can be exploited with a different perspective. Since
fermion lines must be closed, then the contribution of a four fermion operator
to a certain bosonic WC would necessarily lead to a 2-loop diagram, thus being
out of the scope of this analysis. Henceforth, we only need to consider
operators with less than four fermions at the loop. In particular, this means
baryon and lepton number violating operators from the $d_{6}$ set cannot
renormalise the bosonic operators at 1-loop order.

Not all operators are considered for insertions, even if they meet the
requirements above. Among the different classes of operators of $d_{6}$ and
$d_{7}$ we find some of them that are not tree-level generated in the UV
completions of SMEFT [15]. This means that any diagram including these
operators would effectively be loop-suppressed with respect to those that
include only tree-level generated operators. The loop-generated operators
would lead to a process of greater order in loop expansion, which we ignore
for this paper.

One last general remark about the divergent diagrams is in order: in
dimension-8 there are operators containing more than six fields in their
definition (i.e: $\mathcal{O}_{\phi^{8}}$,
$\mathcal{O}^{(1,2,3,4)}_{\phi^{6}}$). For these to be renormalised we would
need to find a diagram with as many legs. It is somewhat easy to reach this
number with certain insertions of operators, for example, with the Weinberg
operator and a $d_{7}$ operator from class $\psi^{2}\phi^{4}$ we get six Higgs
as external fields (diagram 1(d)). Additionally, one can try to reach six or
eight external legs using SM couplings like Yukawas or gauge boson couplings,
however, since we are only considering 1PI diagrams, any vertex we add
increases the number of propagators in the loop, which can render the diagram
finite. Thus, the insertion of operators is capped.

Regarding the dimension-8 Wilson coefficients, all 89+86 physical and
redundant bosonic operators are not renormalised. Recalling the Weinberg
operator

$\mathcal{O}_{\ell\phi}^{(5)}=\epsilon_{ij}\epsilon_{mn}\left(\ell^{i}C\ell^{m}\right)\phi^{j}\phi^{n},$
(2.5)

we see that there are two Higgs field that appear as external legs since the
loop is formed by the fermion lines. Thus, the insertion of
$\alpha_{\ell\phi}$ implies that the final diagram will have at least two
Higgs as external legs. Henceforth, we ignore all those dimension-8 operators
that do not include Higgs fields. Furthermore, considering that for
$d_{5}^{4}$ and $d_{5}^{2}\times d_{6}$ there are two or more insertions of
the Weinberg operator, then diagrams with these insertions do not contribute
to operators with less than four Higgs fields. It so happens that all of the
$d_{7}$ operators meeting the previous properties have at least two Higgs legs
as well, so the diagrams for $d_{5}\times d_{7}$ have at least four Higgs
external legs too. In conclusion, there are no contributions from LNV
insertions to the RGEs of loop-generated bosonic dimension-8 operators.

Finally, we argue why most operators featuring gauge bosons can be neglected
too. All instances containing a gluon field are not renormalised since there
are no gluonic LNV operators. Renormalisation of operators with $B$ bosons is
also heavily restricted due to charge conservation or other symmetries. After
all the above requirements are implemented, there is only one LNV operator
containing $B$ fields: $\mathcal{O}_{\ell\phi B}$. In principle, we could
insert this operator in diagrams like 1(e) or 1(h). However, by definition
$\mathcal{O}_{\ell\phi B}$ is skew-symmetric to flavor transposition and
$\mathcal{O}_{\ell\phi}^{(5)}$ is symmetric. The amplitude with these
insertions would be proportional to a trace of these two operators, which
automatically vanishes. Another possible way for a $B$ field to appear, would
require the terms induced by the covariant derivative acting in the Dirac
Lagrangian: $\overline{\ell^{m}}\not{D}\ell^{m}\supset-
g_{1}Y\overline{\ell^{m}}\not{B}\ell^{m}$. The vertex induced by this
Lagrangian cannot be inserted in diagrams like 1(f) and 1(h) due to charge
conservation. It can, however appear in 1(g) as long as the second gauge
coupling is generated by
$-g_{2}\overline{\ell^{m}}\frac{\sigma^{I}}{2}\not{W}^{I}\ell^{m}$. This
implies that $WB\phi^{4}$ is the only $B$-containing operator that is
renormalised by LNV operators.

Summing up, we have argued the following considerations valid for the purpose
of this paper:

  * •

For all diagrams with insertions of LNV, there is one loop always formed by
fermions.

  * •

Only 2-fermion operators are inserted. In particular, this means the $d_{6}$
LNV and Baryon Number Violating operators cannot renormalise the $d_{8}$
bosonic operators.

  * •

Only tree-level generated operators are inserted.

  * •

There is a limit to the number of yukawa and gauge couplings that can be
inserted in a diagram before it is no longer divergent.

  * •

Operators with less than 4 Higgs do not get any contribution to the RGEs. This
includes class $X^{2}\phi^{2}D^{2}$.

  * •

Operators containing gauge boson fields only be renormalised if there is at
least one $W$.

Following the discussion above, we have simplified considerably the number of
computations needed and the difficulty of the computations by reducing the
scenarios of possible insertions through some general statements.
Nevertheless, we can take this analysis further by focusing on each group of
insertions.

####  2.2.1 Insertions of $d_{5}\times d_{7}$

Let’s study a particular case of $d_{5}\times d_{7}$. If we consider the
dimension-7 Weinberg operator,

$\mathcal{O}_{\ell\phi}^{(7)}=\epsilon_{ij}\epsilon_{mn}\left(\ell^{i}C\ell^{m}\right)\phi^{j}\phi^{n}\left(\phi^{\dagger}\phi\right),$
(2.6)

we could link the $d_{5}$ operator in several different ways. However, since
we are only interested in the bosonic contributions, we need to close all
fermion lines. There is only one way to do this for the purpose of
renormalisation at 1-loop, which is joining the fermion lines of both
operators so that a loop is formed. This way we get a contribution to an
operator with six Higgs (see figure 1(d)).

There are more dimension-7 operators, which could lead to different
contributions, but some of them vanish. This is the case of figure 2(b), that
represents a topology where we could have the insertion of
$\mathcal{O}_{\ell\phi}^{(5)}$ and
$\left(\mathcal{O}_{\ell\phi}^{(7)}\right)^{\dagger}$ or
$\left(\mathcal{O}_{\ell\phi}^{(5)}\right)^{\dagger}$ and
$\mathcal{O}_{\ell\phi}^{(7)}$. However, we find that the total amplitude
vanishes due to cancellation among diagrams which differ from the permutations
of internal (loop) propagators. The presence of the Yukawas is essential for
this cancellation, in particular, diagram 1(d) is very similar to 2(b) but
does not vanish for those insertions. We do not see such cancellations when
the $d_{7}$ operator inserted is from class $\psi^{2}\phi^{2}D^{2}$ since
there is no symmetry in their definitions, so diagrams like 1(i) and 1(g) are
nonzero.

Removing 2(b) implies no direct contribution from $d_{5}\times d_{7}$ appear
in the divergences of the 8-Higgs operator while we can see direct
contributions in operators with less number of Higgs.

####  2.2.2 Insertions of $d_{5}\times d_{5}\times d_{6}$

We could use a similar reasoning for the insertion of two Weinberg operators
and one dimension-6 operator. It is clear by the points argued before that
both $d_{5}$ operators must be in the loop. A $d_{6}$ operator will also be in
the loop if it has fermionic lines, forming the diagrams in Figure 1(b). Note
how there are no physical dimension-6 two-fermion operators without at least
one Higgs field involved, implying we cannot draw a diagram with less than
6-Higgs as external legs, and so, there do not be any direct contribution to
operators with 4-Higgs legs ($i.e.$ classes $\phi^{4}D^{4}$, $X\phi^{4}D^{2}$
and $X^{2}\phi^{4}$).

However, we can also think of insertions of a bosonic $d_{6}$ operators. In
this case, precisely due to the lack of fermions, the vertex cannot be part of
the loop. Instead, it appears in an external leg, thus forming a 1-particle-
reducible diagram. We have chosen not to consider these in the off-shell
approach (as argued in Section 2) so we discard diagrams like 2(c), 2(d) or
2(e). In exchange, we need to use the divergences of the generated redundant
operators to order $1/\Lambda^{2}$ which yield $1/\Lambda^{4}$ terms after the
WC shift.

Once again, there is a remark about the contribution to the 8-Higgs operator,
$\mathcal{O}_{\phi^{8}}$. For the 1PI amplitudes, figure 2(a) shows the
diagram that would need to be considered. Just as 2(b), the total amplitude
for the insertion of $\mathcal{O}_{\ell\phi}^{(5)}$ and $\mathcal{O}_{e\phi}$
vanishes due to mutual cancellation among the contributions from diagrams (by
anti-symmetry of leptonic loop-propagators).

####  2.2.3 Insertions of $d_{5}\times d_{5}\times d_{5}\times d_{5}$

Finally, we arrive at the contribution of four Weinberg operators inserted in
one loop. Since fermion lines must be closed, their WCs will appear contracted
in the same trace, resulting in eight Higgs as external legs (figure 1(a)).
This clearly contributes to the 8-Higgs dimension-8 operator

$\mathcal{O}_{\phi^{8}}=\left(\phi^{\dagger}\phi\right)^{4}\,.$ (2.7)

This amplitude is proportional to a trace of the four WCs. The eight legs of
the diagram are the maximum number allowed for an amplitude of order
$1/\Lambda^{4}$. Since there is only one Weinberg operator, the remaining
freedom resides in the fermions of the loop. Summing all the combinations
yields the only contribution from this insertion. As argued in the previous
sections, the $\phi^{8}$ class of $d_{8}$ operators receives no other
contribution from 1PI diagrams, meaning the lonely term in (3.3) accounts for
all the divergences of $\mathcal{O}_{\phi^{8}}$ coming from the LNV operators
(in the off-shell approach).

####  2.2.4 Renormalisation below dimension-8

Here, we comment on the renormalisation of operators with lower mass
dimension. Since we are only considering contributions to bosonic operators,
it is clear that there must be an even number of LNV operators inserted to get
a non-LNV contribution, which implies that the Wilson coefficients could only
be renormalised to order $1/\Lambda^{2}$ by two insertions of the Weinberg
operator: $d_{5}^{2}$. The bosonic operator coefficients of the renormalisable
SM do not get a contribution from these, and for the $d_{6}$ bosonic operators
are computed in [44].

About the renormalisation to order $1/\Lambda^{4}$ of operators below
dimension-8, we know the divergences carry a factor of $\Lambda^{4}$ in the
amplitude compensated by another mass scale to get the correct mass dimension
below eight for the operators. The only mass scale in the Lagrangian Eq. 2.1
is $\mu_{\phi}$, meaning that a loop of Higgs fields would need to appear in
the diagram. We have already established in section 2.2 that the loop for the
bosonic contributions needs to be formed by fermions, thus no contribution to
order $1/\Lambda^{4}$ is added directly by $(d_{5}^{4})$, $(d_{5}^{2}\times
d_{6})$, $(d_{5}\times d_{7})$ to the RGEs of Wilson coefficients from bosonic
operators with mass dimension below 8. Nonetheless, this does not necessarily
imply that there do not be any contribution. The argument made here relies on
the fact that we only consider 1PI diagrams, thus, diagrams that are not 1PI
may actually be valid and relevant. In the off-shell approach, these
contribution is encoded in the non-physical operators from the Green’s basis.
Once that the divergences have been computed, one must shift the physical
coefficients as specified in [41], this is a consequence of applying the
Equations of Motion. The shift causes a mixing of operators of different
classes with extra couplings which can and in fact gives an indirect nonzero
contribution to the RGE of some operators even if all their divergent 1PI
diagrams are null.

##  3 The RGEs

In this section, we present the RGEs of the dimension-8 Wilson coefficients up
to order $1/\Lambda^{4}$ and discuss the derivation in detail. The main
results have been uploaded to a public Mathematica notebook.

In section 3.1, we show the divergences we use to get the RGEs in 3.2, where
we also tabulated the greatest contributions to the anomalous dimension
matrix. The process is simple, but we explain it here for a complete analysis.

In general, we shall have three sources of LNV contributions to the anomalous
dimension matrix to order $1/\Lambda^{4}$:

$16\pi^{2}\frac{\rm{d}}{\rm{d}\ln{\mu}}c_{k}(\mu):=\dot{c}_{k}=\gamma_{k}\alpha_{\ell\phi}\alpha_{\ell\phi}\alpha_{\ell\phi}\alpha_{\ell\phi}+\gamma_{ik}\alpha_{\ell\phi}\alpha_{\ell\phi}\beta_{i}+\gamma_{jk}\alpha_{\ell\phi}\omega_{j}\,,$
(3.1)

where hereafter $c_{k}$ are Wilson coefficients of dimension-8 bosonic
operators (shown in Table 4) but any other coefficient of dimension lower than
eight can be either bosonic or fermionic (see Table 5). Note this only affects
$\beta_{i}$, which are the dimension-6 Wilson coefficients – the dimension-5
($\alpha_{\ell\phi}$) and dimension-7 ($\omega_{j}$) only include fermionic
operators.

Following the steps in section 2, the divergences are obtained. Now, they are
applied to the formula for the computation of the RGEs up to 1-loop order:

$\dot{c}_{i}=-c_{i}\sum_{j}n_{j}x_{j}\frac{\partial}{\partial
x_{j}}\left(\frac{\tilde{c}_{i}}{c_{i}}\right),$ (3.2)

where, $n_{j}$ is the tree-level anomalous dimension***For an operator with
$n_{\psi}$ fermions, $n_{X}$ field strength tensors, $n_{\phi}$ Higgs fields
then $n_{j}=n_{\psi}+n_{X}+n_{\phi}-2$ in $D=4-2\epsilon$., and $x_{j}$ are
all the Wilson coefficients (running or not) that enter the Lagrangian.
$c_{i}$ are the Wilson coefficients that are renormalised, whereas,
$\tilde{c}_{i}$ are the computed divergences. They should also include
contributions from the field redefinition needed to canonically normalize the
Higgs kinetic term as well as the Wave Function Renormalisation (WFR) for
gauge bosons, however these only affect self-contributions to the RGEs, which
we do not consider here.

For an exhaustive computation of the anomalous dimension matrix to order
$1/\Lambda^{4}$, we must also consider the divergences of $d_{6}$ operators to
order $1/\Lambda^{2}$ by LNV operators. The insertions can only be of two
$d_{5}$ operators. Nonetheless, we have attached these results in section 3.1
for completeness.

###  3.1 Divergences

The calculation of the divergences is the key step to determine the running of
the Wilson coefficients. We have already shown in Section 2 the 1PI diagrams
whose amplitudes we need to compute for renormalisation. These divergent
amplitudes are matched with those of the corresponding tree-level processes
with dimension-8 vertices.

To do this, both the SMEFT and the dimension-8 effective Lagrangians are
implemented in FeynRules [45] models and the amplitudes are computed with the
help of FormCalc [46] and FeynArts [47]. The amplitudes are a sum of
expressions that depend on the couplings and kinematic invariants (i.e. a set
of algebraically independent contractions of momenta and polarisation
vectors). Equating the amplitudes for each process, we get a set of as many
equations as unique kinematic invariants. The solutions of these equations are
the divergences for the dimension-8 Wilson coefficients of the Green’s basis
operators, which we present here.

$\displaystyle\tilde{c}_{\phi^{8}}$
$\displaystyle\supset-\frac{1}{2\pi^{2}\epsilon}\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right],$
(3.3) $\displaystyle\tilde{c}_{\phi^{6}D^{2}}^{(1)}$
$\displaystyle\supset\frac{1}{2\pi^{2}\epsilon}\text{Tr}\left[-3\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}+4\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}\right]+\frac{1}{4\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}y^{e}\omega_{\ell\phi
De}^{*}\right]\right)$
$\displaystyle+\frac{1}{2\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi}^{\dagger}\right]\right)-\frac{1}{16\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\omega_{\ell\phi
D}^{(2)}\right]\right)$
$\displaystyle-\frac{1}{16\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\left(\omega_{\ell\phi
D}^{(1)}+\left(\omega_{\ell\phi D}^{(1)}\right)^{T}\right)\right]\right),$
(3.4) $\displaystyle\tilde{c}_{\phi^{6}D^{2}}^{(2)}$
$\displaystyle\supset\frac{1}{2\pi^{2}\epsilon}\text{Tr}\left[-2\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}+\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}\right]-\frac{1}{4\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}y^{e}\omega_{\ell\phi
De}^{*}\right]\right)$
$\displaystyle+\frac{1}{4\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi}^{\dagger}\right]\right)-\frac{1}{16\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\omega_{\ell\phi
D}^{(2)}\right]\right)$
$\displaystyle-\frac{1}{16\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\left(\omega_{\ell\phi
D}^{(1)}+\left(\omega_{\ell\phi D}^{(1)}\right)^{T}\right)\right]\right),$
(3.5) $\displaystyle\tilde{c}_{\phi^{6}D^{2}}^{(3)}$
$\displaystyle\supset\frac{1}{4\pi^{2}\epsilon}\text{Tr}\left[-\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}+\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}\right]-\frac{1}{8\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi}^{\dagger}\right]\right),$
(3.6) $\displaystyle\tilde{c}_{\phi^{6}D^{2}}^{(4)}$
$\displaystyle\supset\frac{1}{8\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi}^{\dagger}\right]\right),$
(3.7) $\displaystyle\tilde{c}_{\phi^{4}D^{4}}^{(2)}$
$\displaystyle\supset\frac{1}{4\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.8) $\displaystyle\tilde{c}_{\phi^{4}D^{4}}^{(10)}$
$\displaystyle\supset\frac{1}{8\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.9) $\displaystyle\tilde{c}_{\phi^{4}D^{4}}^{(11)}$
$\displaystyle\supset\frac{1}{8\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.10)
$\displaystyle\tilde{c}_{\phi^{4}D^{4}}^{(12)}$
$\displaystyle\supset\frac{1}{4\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.11)
$\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(1)}$
$\displaystyle\supset\frac{g_{2}}{12\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\alpha_{L\phi D}^{(2)}\right]\right),$
(3.12) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(2)}$
$\displaystyle\supset\frac{g_{2}}{8\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.13) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(3)}$
$\displaystyle\supset\frac{g_{2}}{16\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.14) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(4)}$
$\displaystyle\supset-\frac{g_{2}}{16\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.15) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(5)}$
$\displaystyle\supset\frac{g_{2}}{48\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.16) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(6)}$
$\displaystyle\supset\frac{g_{2}}{48\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.17) $\displaystyle\tilde{c}_{W\phi^{4}D^{2}}^{(7)}$
$\displaystyle\supset\frac{g_{2}}{48\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.18) $\displaystyle\tilde{c}_{W^{2}\phi^{4}}^{(1)}$
$\displaystyle\supset\frac{g_{2}^{2}}{192\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)-\frac{g_{2}}{8\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.19)
$\displaystyle\tilde{c}_{W^{2}\phi^{4}}^{(2)}$
$\displaystyle\supset\frac{g_{2}^{2}}{128\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)+\frac{g_{2}}{8\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.20)
$\displaystyle\tilde{c}_{W^{2}\phi^{4}}^{(3)}$
$\displaystyle\supset\frac{g_{2}^{2}}{192\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)+\frac{g_{2}}{8\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.21)
$\displaystyle\tilde{c}_{W^{2}\phi^{4}}^{(4)}$
$\displaystyle\supset\frac{g_{2}^{2}}{128\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)-\frac{g_{2}}{8\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.22)
$\displaystyle\tilde{c}_{WB\phi^{4}}^{(1)}$
$\displaystyle\supset\frac{g_{1}g_{2}}{96\pi^{2}\epsilon}\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.23) $\displaystyle\tilde{c}_{WB\phi^{4}}^{(2)}$
$\displaystyle\supset\frac{g_{1}g_{2}}{64\pi^{2}\epsilon}\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right).$
(3.24)

The absent coefficients are not receiving any contribution from LNV operators.

###  3.2 Explicit expression for the RGEs

We compute the terms of anomalous dimension matrix as discussed at the
beginning of this section. The nonzero RGEs for the dimension-8 physical basis
operators are:

$\displaystyle\dot{c}_{\phi^{8}}$ $\displaystyle=8\lambda_{\phi}\beta_{\phi
D}\,\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]+32\lambda_{\phi}\,\text{Tr}\left[-\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}+\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]$
$\displaystyle-16\lambda_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi}\right]\right)-\lambda_{\phi}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right)$
$\displaystyle+16\,\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right],$
(3.25) $\displaystyle\dot{c}_{\phi^{6}}^{(1)}$ $\displaystyle=4\beta_{\phi
D}\,\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]+16\,\text{Tr}\left[3\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}-4\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]$
$\displaystyle-16\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi}\right]\right)+\frac{7}{3}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right)$
$\displaystyle-32\lambda_{\phi}\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)-4\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}y^{e}\omega_{\ell\phi
De}^{*}\right]\right)$
$\displaystyle-2\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\left(\omega_{\ell\phi
D}^{(1)}+\left(\omega_{\ell\phi D}^{(1)}\right)^{T}\right)\right]\right)$
$\displaystyle-2\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.26) $\displaystyle\dot{c}_{\phi^{6}}^{(2)}$
$\displaystyle=8\beta_{\phi
D}\,\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]+16\,\text{Tr}\left[2\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}-\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]$
$\displaystyle-8\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi}\right]\right)+\frac{1}{12}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi D}^{(2)}\right]\right)$
$\displaystyle-16\lambda_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\alpha_{\ell\phi
D}^{(2)}\right]\right)+4\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\left(y^{e}\right)\omega_{\ell\phi
De}^{*}\right]\right)$
$\displaystyle+2\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\left(y^{e}\right)^{*}\left(y^{e}\right)\left(\omega_{\ell\phi
D}^{(1)}+\left(\omega_{\ell\phi D}^{(1)}\right)^{T}\right)\right]\right)$
$\displaystyle+2\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\left(y^{e}\right)^{*}\left(y^{e}\right)\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.27) $\displaystyle\dot{c}_{\phi^{4}}^{(2)}$
$\displaystyle=-8\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.28) $\displaystyle\dot{c}_{W\phi^{4}D^{2}}^{(1)}$
$\displaystyle=-4g_{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.29) $\displaystyle\dot{c}_{W\phi^{4}D^{2}}^{(2)}$
$\displaystyle=-4g_{2}\,\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.30) $\displaystyle\dot{c}_{W\phi^{4}D^{2}}^{(3)}$
$\displaystyle=-2g_{2}\,\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.31) $\displaystyle\dot{c}_{W\phi^{4}D^{2}}^{(4)}$
$\displaystyle=2g_{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi D}^{(2)}\right]\right),$
(3.32) $\displaystyle\dot{c}_{W^{2}\phi^{4}}^{(1)}$
$\displaystyle=-\frac{1}{4}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)+4g_{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.33)
$\displaystyle\dot{c}_{W^{2}\phi^{4}}^{(2)}$
$\displaystyle=-\frac{1}{4}g_{2}^{2}\,\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)-4g_{2}\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.34)
$\displaystyle\dot{c}_{W^{2}\phi^{4}}^{(3)}$
$\displaystyle=-\frac{1}{4}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)-4g_{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.35)
$\displaystyle\dot{c}_{W^{2}\phi^{4}}^{(4)}$
$\displaystyle=-\frac{1}{4}g_{2}^{2}\,\text{Im}\left(\text{Tr}\left[2\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}+\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)+4g_{2}\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}\omega_{\ell\phi
W}^{\dagger}\right]\right),$ (3.36) $\displaystyle\dot{c}_{WB\phi^{4}}^{(1)}$
$\displaystyle=-g_{1}g_{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}\right]\right)-\frac{1}{2}g_{1}g_{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.37) $\displaystyle\dot{c}_{WB\phi^{4}}^{(2)}$
$\displaystyle=-g_{1}g_{2}\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}\right]\right)-\frac{1}{2}g_{1}g_{2}\,\text{Im}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right).$ (3.38)

As we discussed in section 2.2.4, there is no renormalisation at order
$1/\Lambda^{4}$ coming directly from the 1PI diagrams with LNV vertices for
any coefficient of dimension less than eight. However, the redundancies of the
operators in the Green’s basis do contribute to these equations. All of these
yield a non-vanishing RGE for some coefficients. Again, we only show the
nonzero RGEs:

$\displaystyle\dot{\lambda}_{\phi}$
$\displaystyle=-8\mu^{4}_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.39) $\displaystyle\dot{c}_{\phi}$
$\displaystyle=-4\mu^{2}_{\phi}\beta_{\phi
D}\,\text{Tr}\left[\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]+16\mu^{2}_{\phi}\,\text{Tr}\left[\beta_{\phi\ell}^{(1)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]$
$\displaystyle-16\mu^{2}_{\phi}\,\text{Tr}\left[\beta_{\phi\ell}^{(3)}\alpha_{\ell\phi}\alpha_{\ell\phi}^{\dagger}\right]+8\mu^{2}_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi}\right]\right)$
$\displaystyle+\mu^{2}_{\phi}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(1)}\right]\right)+\frac{1}{2}\mu^{2}_{\phi}g_{2}^{2}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$
$\displaystyle+16\lambda_{\phi}\mu^{2}_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)$ (3.40) $\displaystyle\dot{c}_{\phi\square}$
$\displaystyle=16\mu^{2}_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (3.41) $\displaystyle\dot{c}_{\phi D}$
$\displaystyle=16\mu^{2}_{\phi}\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right).$ (3.42)

All of the previous results for the RGEs are expressed in table 2. There it is
easy to check the numerical contribution for each operator coming from all the
possible sources detailed previously. In cases where there is more than one
term for the same contribution, we show only the greatest. Naive loop
suppression $\gamma\sim 1$ competes with a high numerical coefficient for some
of the terms, specially those of class $\phi^{6}D^{2}$, which hold the highest
terms in all cases.

| $\left(\alpha_{\ell\phi}\right)^{4}$ | $\left(\alpha_{\ell\phi}\right)^{2}\beta_{\phi D}$ | $\left(\alpha_{\ell\phi}\right)^{2}\beta_{\phi\ell}^{(1)}$ | $\left(\alpha_{\ell\phi}\right)^{2}\beta_{\phi\ell}^{(3)}$ | $\alpha_{\ell\phi}\omega_{\ell\phi}$ | $\alpha_{\ell\phi}\omega_{\ell\phi D}^{(1)}$ | $\alpha_{\ell\phi}\omega_{\ell\phi D}^{(2)}$ | $\alpha_{\ell\phi}\omega_{\ell\phi De}$ | $\alpha_{\ell\phi}\omega_{\ell\phi W}$
---|---|---|---|---|---|---|---|---|---
$\gamma_{\phi^{8}}$ | 16 | $8\lambda_{\phi}$ | $32\lambda_{\phi}$ | $32\lambda_{\phi}$ | $16\lambda_{\phi}$ | $2\lambda_{\phi}g_{2}^{2}$ | $\lambda_{\phi}g_{2}^{2}$ | 0 | 0
$\gamma_{\phi^{6}}^{(1)}$ | 0 | 4 | 48 | 64 | 16 | $\frac{14}{3}g_{2}^{2}$ | $32\lambda_{\phi}$ | $4y^{e}$ | 0
$\gamma_{\phi^{6}}^{(2)}$ | 0 | 8 | 32 | 16 | 8 | $\frac{1}{6}g_{2}^{2}$ | $16\lambda_{\phi}$ | $4y^{e}$ | 0
$\gamma_{\phi^{4}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $8$ | 0 | 0
$\gamma_{W\phi^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | $8g_{2}$ | $4g_{2}$ | 0 | 0
$\gamma_{W\phi^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | $8g_{2}$ | $4g_{2}$ | 0 | 0
$\gamma_{W\phi^{4}D^{2}}^{(3)}$ | 0 | 0 | 0 | 0 | 0 | $4g_{2}$ | $2g_{2}$ | 0 | 0
$\gamma_{W\phi^{4}D^{2}}^{(4)}$ | 0 | 0 | 0 | 0 | 0 | $4g_{2}$ | $2g_{2}$ | 0 | 0
$\gamma_{W^{2}\phi^{4}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}g_{2}^{2}$ | $\frac{1}{4}g_{2}^{2}$ | 0 | $4g_{2}$
$\gamma_{W^{2}\phi^{4}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}g_{2}^{2}$ | $\frac{1}{4}g_{2}^{2}$ | 0 | $4g_{2}$
$\gamma_{W^{2}\phi^{4}}^{(3)}$ | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}g_{2}^{2}$ | $\frac{1}{4}g_{2}^{2}$ | 0 | $4g_{2}$
$\gamma_{W^{2}\phi^{4}}^{(4)}$ | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}g_{2}^{2}$ | $\frac{1}{4}g_{2}^{2}$ | 0 | $4g_{2}$
$\gamma_{WB\phi^{4}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | $g_{1}g_{2}$ | $\frac{1}{2}g_{1}g_{2}$ | 0 | 0
$\gamma_{WB\phi^{4}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | $g_{1}g_{2}$ | $\frac{1}{2}g_{1}g_{2}$ | 0 | 0
$\gamma_{\phi}$ | 0 | $4\mu^{2}_{\phi}$ | $16\mu^{2}_{\phi}$ | $16\mu^{2}_{\phi}$ | $8\mu^{2}_{\phi}$ | $\mu^{2}_{\phi}g_{2}^{2}$ | $16\mu^{2}_{\phi}\lambda_{\phi}$ | 0 | 0
$\gamma_{\phi\square}$ | 0 | 0 | 0 | 0 | 0 | 0 | $16\mu^{2}_{\phi}$ | 0 | 0
$\gamma_{\phi D}$ | 0 | 0 | 0 | 0 | 0 | 0 | $16\mu^{2}_{\phi}$ | 0 | 0
$\gamma_{\lambda_{\phi}}$ | 0 | 0 | 0 | 0 | 0 | 0 | $8\mu^{4}_{\phi}$ | 0 | 0

Table 2: Anomalous dimension matrix. The columns represent the greatest terms
from each contribution in absolute value that renormalise the coefficients in
the rows. See equations 3.2 – 3.42 for complete RGEs.

###  3.3 Detailed example

Let us work out a detailed example of this calculation for clarity: the
contribution of all the LNV operators to the RGE of $c_{\phi^{4}}^{(2)}$.
First of all, we implement the Lagrangian (2.1) in a FeynRules model. With
this, we generate the topologies and processes in FeynArts, which paired with
FormCalc computes the amplitudes for the different processes. In particular,
for $c_{\phi^{4}}^{(2)}$ we get information from the process
$\phi^{0}\rightarrow\phi^{0}\phi^{+}\phi^{-}$. Of course, this is not the only
process that could yield the divergence for this particular coefficient.
Generally, different processes give the divergences for the same set of
coefficients, so it is a matter of finding a suitable process in each case. In
our example we have:

$\displaystyle-\text{i}\mathcal{A}_{IR}$
$\displaystyle=2c_{\phi^{4}}^{(1)}\left(-\kappa_{1213}+\kappa_{1322}+\kappa_{1323}\right)+2c_{\phi^{4}}^{(2)}\left(-\kappa_{1123}+\kappa_{1223}+\kappa_{1323}\right)$
$\displaystyle+2c_{\phi^{4}}^{(3)}\left(-\kappa_{1213}+\kappa_{1223}+\kappa_{1233}\right)+2c_{\phi^{4}}^{(4)}\left(\kappa_{1212}+\kappa_{1213}-\kappa_{1223}-\kappa_{1233}\right)$
$\displaystyle+c_{\phi^{4}}^{(4)}\left(-\kappa_{1112}+\kappa_{1113}-\kappa_{1123}-\kappa_{1133}-\kappa_{1222}+\kappa_{1322}-\kappa_{2223}-\kappa_{2233}\right)$
$\displaystyle+c_{\phi^{4}}^{(6)}\left(\kappa_{1112}-\kappa_{1113}-\kappa_{1122}-\kappa_{1123}+\kappa_{1233}-\kappa_{1333}-\kappa_{2223}-\kappa_{2333}\right)$
$\displaystyle+2c_{\phi^{4}}^{(6)}\left(\kappa_{1313}+\kappa_{1213}-\kappa_{1322}-\kappa_{1323}\right)+4c_{\phi^{4}}^{(8)}\left(-\kappa_{1112}-\kappa_{1113}+\kappa_{1123}\right)$
$\displaystyle+2c_{\phi^{4}}^{(8)}\left(\kappa_{1111}+\kappa_{1122}+\kappa_{1133}+\kappa_{2233}\right)+c_{\phi^{4}}^{(10)}\left(\kappa_{3333}+\kappa_{1122}+\kappa_{1133}+\kappa_{2233}\right)$
$\displaystyle+2c_{\phi^{4}}^{(10)}\left(-\kappa_{1233}-\kappa_{1333}+\kappa_{2333}\right)+c_{\phi^{4}}^{(11)}\left(\kappa_{2222}+\kappa_{1122}+\kappa_{1133}+\kappa_{2333}\right)$
$\displaystyle+2c_{\phi^{4}}^{(11)}\left(-\kappa_{1222}-\kappa_{1322}+\kappa_{2223}\right)+2c_{\phi^{4}}^{(12)}\left(\kappa_{2323}+\kappa_{1123}-\kappa_{1223}-\kappa_{1323}\right)$
$\displaystyle+c_{\phi^{4}}^{(12)}\left(-\kappa_{1122}-\kappa_{1133}+\kappa_{1222}+\kappa_{1233}+\kappa_{1322}+\kappa_{1333}+\kappa_{2223}+\kappa_{2333}\right),$
(3.43)

where $\kappa_{ijkl}=p_{i}\cdot p_{j}p_{k}\cdot p_{l}$ are the kinematic
invariants and we are not including CP-odd terms since we already know they do
not contribute to the CP-even $c_{\phi^{4}}^{(2)}$. This amplitude has been
built at tree level including redundant operators in the dimension-8 Green’s
basis so that only 1PI diagrams are considered here, as explained in section
2. Consequently with this method, the amplitude is off-shell. Now for the
1-loop amplitude we have:

$-\text{i}\mathcal{A}_{UV}=\frac{1}{8\pi^{2}\epsilon}\text{Re}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\left(\kappa_{2222}+\kappa_{3333}+2\kappa_{2233}+4\kappa_{2323}+4\kappa_{2333}\right),$
(3.44)

which indicates that the only contribution for this process comes from the
4-Higgs diagram displayed in Figure 1(c).

Equating the two amplitudes effectively matches the Wilson coefficients to
their divergences. The nonzero solutions for our example are:

$c_{\phi^{4}}^{(2)}=2c_{\phi^{4}}^{(10)}=2c_{\phi^{4}}^{(11)}=c_{\phi^{4}}^{(12)}=\frac{1}{4\pi^{2}\epsilon}\text{Re}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\,.$ (3.45)

Now we have obtained the divergences of $c_{\phi^{4}}^{(2)}$, but following
the off-shell method, we also need the divergences for the redundant
coefficients that appear in the redefinition of $c_{\phi^{4}}^{(2)}$.
Following Ref. [41], the shift for this coefficient is:

$c_{\phi^{4}}^{(2)}\rightarrow
c_{\phi^{4}}^{(2)}+g_{1}c_{B\phi^{2}D^{4}}^{(3)}+g_{2}c_{W\phi^{2}D^{4}}^{(3)}-g_{1}^{2}c_{B^{2}D^{4}}-g_{2}^{2}c_{W^{2}D^{4}}.$
(3.46)

Now, applying all the divergences to Eq. 3.46, we get the physical 1-loop
coefficient:

$\tilde{c}_{\phi^{4}}^{(2)}=\frac{1}{4\pi^{2}\epsilon}\text{Re}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\,.$ (3.47)

Now, according to equation Eq. 3.2, we need the tree-level anomalous dimension
$n_{j}$ for operators $\alpha_{\ell\phi}$ and $\omega_{\ell\phi D}^{(2)}$. One
can trivially check that

$\text{Mass
dim.}\left[\mu^{-(n_{\ell\phi}^{(5)})\epsilon}\frac{\alpha_{\ell\phi}}{\Lambda}\mathcal{O}_{\ell\phi}^{(5)}\right]\mapsto
4-(n_{\ell\phi}^{(5)})\epsilon=4-2\epsilon\Rightarrow n_{\ell\phi}^{(5)}=2.$
(3.48)

And, with the same proceeding $n_{\ell\phi D}^{(2)}=3$. Finally, all that
remains is to use the definition of the RGEs Eq. 3.2, which is straightforward
since we have all the ingredients needed. Note that $x_{j}$ in the formula
comprehends all the coefficients, running or not, which includes all the WCs,
but also the SM couplings and masses $\mu^{2}_{\phi}$, $\lambda_{\phi}$,
$g_{1}$, $g_{2}$, and $g_{3}$. In this case the derivative is trivial and so,
the RGE reads:

$\dot{c}_{\phi^{4}}^{(2)}=-8\,\text{Re}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\,.$ (3.49)

Of course, one could add more contributions to Eq. 3.47 in order to have a
complete RGE to order $1/\Lambda^{4}$, but in this paper we only focus in
those involving LNV operators, the rest of the terms (namely those coming from
other dimension-8 operators or from pairs of dimension-6) are explicitly shown
in references [27] and [28].

##  4 Discussions and outlook

Here, we discuss some implications on BSM parameter space based on the basis
of these RGEs.

###  4.1 Positivity bounds

Positivity bounds are restrictions on S-matrix elements derived using the
standard axioms – analyticity, unitarity, and crossing symmetry on the
underlying theory. In the context of SMEFT, these bounds constrain the space
of Wilson coefficients of dimension-8 operators. The positivity bounds for
$2\rightarrow 2$ scattering amplitude $\mathcal{A}(s,t)$ in the forward
scattering limit $(t=0)$ are deduced using [48],

$\displaystyle\left.\frac{d^{2}}{ds^{2}}\mathcal{A}(s,0)\right|_{s=0}\geq 0.$
(4.1)

Assuming that $A(s,0)$ is analytic around the origin and is expanded around
$s=0$,

$\displaystyle A(s,0)=a_{0}+a_{1}s+a_{2}s^{2}+\ldots\,,$ (4.2)

which implies that $a_{2}\geq 0$ following Eq. 4.1. $a_{2}$ is defined by
combinations of dimension-8 operators (or pairs of dimension-6 operators,
etc.), and these combinations are restricted by the positivity bounds. For
example, considering the process $\phi\phi\rightarrow\phi\phi$ and translating
the positivity bounds on the Wilson coefficients [32],

$\displaystyle\phi_{1}\phi_{2}\rightarrow\phi_{1}\phi_{2}$
$\displaystyle:c^{(2)}_{\phi^{4}}\geq 0\ ,$
$\displaystyle\phi_{1}\phi_{3}\rightarrow\phi_{1}\phi_{3}$
$\displaystyle:c^{(1)}_{\phi^{4}}+c^{(2)}_{\phi^{4}}\geq 0\ ,$ (4.3)
$\displaystyle\phi_{1}\phi_{1}\rightarrow\phi_{1}\phi_{1}$
$\displaystyle:c^{(1)}_{\phi^{4}}+c^{(2)}_{\phi^{4}}+c^{(3)}_{\phi^{4}}\geq 0\
.$

The amplitude around $s=0$ can be irregular in certain cases, such as, when
contributions from loops carrying massless propagators. In such cases there
are branch cuts extending to the origin. Generally, one arrives at Eq 4.1 by
lending a small mass ‘$m$’ to the massless state to regularise the singularity
at the origin, and resetting it to zero at a later stage. But, one needs to
scrutinize the deduced conclusion when there are loops with massless
propagators present [36].

For $\phi^{4}D^{4}$ operators, the running effects of lower dimensional
operators ($\phi^{4}$ and $\phi^{4}D^{2}$) dominates the corresponding
amplitude in the limit $m\rightarrow 0$ [36]. The positivity bounds on
$\phi^{4}D^{4}$ class are respected by the running of effective interactions,
which implies provided that these $d_{8}$ WCs are absent at the tree-level,
the $d_{8}$ RGEs restrict arbitrary values of the WCs. For instance, consider
the bound $c_{\phi^{4}}^{(2)}(\Lambda)\geq 0$, and,

$\displaystyle\dot{c}_{\phi^{4}}^{(2)}=-8\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right),$ (4.4)

from Eq. 3.28, then, Eq. 3.1 evaluates to,

$\displaystyle
16\pi^{2}c_{\phi^{4}}^{(2)}(\mu)=8\,\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)\text{ln}\frac{\Lambda}{\mu}$ $\displaystyle\geq 0$
$\displaystyle\Rightarrow\ \
\text{Re}\left(\text{Tr}\left[\alpha_{\ell\phi}^{\dagger}\omega_{\ell\phi
D}^{(2)}\right]\right)$ $\displaystyle\geq 0,$ (4.5)

for $\mu<\Lambda$ in the limit of scale-invariance of the LNVs. This
inequality imposes significant restrictions on new physics parameter space.
For instance, assuming only one flavor index and real valued WCs,
$\alpha_{\ell\phi,mm}$ and $\omega_{\ell\phi D,mm}^{(2)}$ must have same signs
in any UV completion of SMEFT that does not generate $\phi^{4}D^{4}$ WCs at
tree level, whereas for three flavor indices,

$\displaystyle\sum_{mn}^{3}\alpha_{\ell\phi,mn}\omega_{\ell\phi
D,mn}^{(2)}\geq 0.$ (4.6)

We highlight the fact that this restriction is deduced purely using IR
information $i.e.$ the RGEs calculation in the IR (SMEFT in our case) and the
$d_{8}$ WC positivity bounds directly impose constrains on the UV space.

We inspect this inequality in BSMs where these LNVs are generated at tree-
level matching, but $\phi^{4}D^{4}$ class of $d_{8}$ operators do not. A
suitable choice for such BSMs is Type-I and III seesaw models.

We add two heavy multiplets to SM particle content
$N^{i}:(1,1,0)_{\frac{1}{2}}$ and $\Sigma^{i}:(1,3,0)_{\frac{1}{2}}$ with SM
gauge and spin quantum numbers shown as: $(3_{C},2_{L},1_{Y})_{S}$ . The BSM
Lagrangian is defined as,

$\displaystyle\mathcal{L}=$ $\displaystyle\
\mathcal{L}_{\text{SM}}+\frac{1}{2}\bar{N}(i\not{\partial}-M_{N})N+\frac{1}{2}\text{Tr}[\bar{\Sigma}(i\not{D}-M_{\Sigma})\Sigma]$
$\displaystyle-\frac{1}{2}\left\\{Y^{N\dagger}_{ij}\,\bar{N}^{i}\,\widetilde{\phi}^{\dagger}\,\ell^{j}+(Y^{N})^{T}_{ij}\,\bar{N}^{i}\,\widetilde{\phi}^{T}\,\ell^{Cj}+Y^{\Sigma}_{ij}\,\widetilde{\phi}^{\dagger}\bar{\Sigma}^{i}\,\ell^{j}+Y^{\Sigma\ast}_{ij}\,\phi^{\dagger}\bar{\Sigma}^{i}\,i\sigma_{2}\ell^{Cj}\text{+
h.c.}\right\\},$ (4.7)

where, $\Sigma$ is a $SU(2)$ triplet and is in 2-dimensional matrix form. We
integrate out the heavy fields at tree-level on to SMEFT up to dimension-7,

$\displaystyle\mathcal{O}_{7ij}^{\prime}$
$\displaystyle=(\bar{\ell}^{C}_{i}\tilde{\phi}^{\ast})\not{D}^{2}(\tilde{\phi}^{\dagger}\ell_{j})\
,\
\mathcal{C}_{7ij}^{\prime}=-\frac{\left[(Y^{N})^{\ast}(Y^{N})^{\dagger}\right]_{ij}}{2M_{N}^{3}}-\frac{\left[(Y^{\Sigma})^{T}(Y^{\Sigma})\right]_{ij}}{2M_{\Sigma}^{3}}\,,$
(4.8)

$\mathcal{O}^{\prime}_{7ij}$ is a dimension-7 operator, and is related to
operators in the basis we have considered (table 5). By using the SM equation
of motions and properties of Dirac-gamma matrices, one can reduce it to
operators of classes $\psi^{2}\phi^{2}$, $\psi^{2}\phi^{4}$, $\psi^{4}\phi$,
$\psi^{2}\phi^{3}D$, and $\psi^{2}\phi^{2}X$. Note that at $d_{7}$ only
$\mathcal{O}_{\ell\phi D}^{(2)}$ (class $\psi^{2}\phi^{2}D^{2}$) renormalises
the $\phi^{4}D^{4}$ operators. For these models, $\mathcal{O}_{\ell\phi
D}^{(2)}$ is not generated from tree-level matching, and so, $\omega_{\ell\phi
D}^{(2)}=0$. Therefore, we conclude that Eq. 4.1 is satisfied in Type-I and
III seesaw models.

###  4.2 Constraints from T-parameter

The Wilson coefficients $\beta_{\phi D}$ (dimension-6) and
$c^{(2)}_{\phi^{6}}$ (dimension-8) are restricted from the bounds on
T-parameter [49],

$\displaystyle T=-\frac{1}{2\alpha}\frac{v^{2}}{\Lambda^{2}}\left(\beta_{\phi
D}+c^{(2)}_{\phi^{6}}\frac{v^{2}}{\Lambda^{2}}\right),$ (4.9)

where, $\alpha\sim\frac{1}{137}$ is the fine structure constant. The LNVs
renormalise $\mathcal{O}^{(2)}_{\phi^{6}}$ (Eq. 3.27), and thus receive
resultant constraint from T-parameter. This becomes important where
$\phi^{4}D^{2}$ and $\phi^{6}D^{2}$ classes do not appear at tree-level. In
such case, from the RGE of $c^{(2)}_{\phi^{6}}$ in the limit of scale
invariant WCs,

$\displaystyle
T=-\frac{1}{4\pi^{2}\alpha}\frac{v^{4}}{\Lambda^{4}}\ln\left[\frac{\Lambda}{\mu}\right]\sum_{mn}^{3}\alpha_{\ell\phi,mn}\omega_{\ell\phi,mn},$
(4.10)

assuming that the WCs of only Weinberg-like operators ($\alpha_{\ell\phi}$ and
$\omega_{\ell\phi}$) are non-vanishing and real valued. This put restrictions
on the $\alpha_{\ell\phi}-\omega_{\ell\phi}$ plane, and this can be compared
to the bounds from neutrino masses [50],

$\displaystyle\left(M_{N}\right)_{mn}=-\frac{v^{2}}{\Lambda}\left(\alpha_{\ell\phi,mn}+\frac{v^{2}}{2\Lambda^{2}}\omega_{\ell\phi,mn}\right).$
(4.11)

It is straightfoward to visualize the restrictions on $\alpha_{\ell\phi}$ vs
$\omega_{\ell\phi}$ plane taking both T-parameter and neutrino mass bounds in
the case of single non-vanishing flavor direction as shown in Fig. 3. Note
that, in general, neutrino masses do not restrain $\alpha_{\ell\phi}$ and
$\omega_{\ell\phi}$ to possess large values. This blind direction is lifted by
the T-parameter, as it imposes an upper bound to these WCs on the account of
the RGEs of dimension-8 operators, which is shown in the figure.

Figure 3: Displaying the restrictions imposed on Weinberg-like operators,
$\mathcal{O}_{\ell\phi}^{(5)}$ at dimension-5 and
$\mathcal{O}_{\ell\phi}^{(7)}$ at dimension-7, translated from bounds on
T-parameter for one non-vanishing flavor direction. Neutrino mass bounds
restrict these two Wilson coefficients to lie on the shown blue contour,
whereas T-parameter bound restricts the Wilson coefficients to lie on the
yellow region. The plot is created for $\Lambda=1$ TeV, and $\mu=246$ GeV,
with inputs $T=0.10\pm 0.12$ [49], and $M_{N}<0.081$ eV [50].

##  5 Conclusions

Throughout the last years and with the effort of many authors, the different
contributions to the running of the Wilson coefficients of the SMEFT have been
studied and revealed part by part. With Eq. 3.2 – Eq. 3.42, we have considered
in this paper for the first time the last set of operators yet-not-examined
for the running of bosonic operators at order $1/\Lambda^{4}$. Now that the
LNV operators are included in the RGEs, all the possible contributions to
bosonic operators of the SMEFT are known explicitly. There are some general
remarks about the anomalous dimension matrices of the operators considered
here that also apply in this case and are worth mentioning.

All the bosonic operators that are being renormalised arise at tree-level in
the UV-completions of SMEFT. All the bosonic operators that are loop-generated
(have less than four Higgs fields) are not renormalised for the reasons
mentioned in Section 2.2. We have eliminated all the vanishing divergences due
to irrelevant or trivial amplitudes (for example, due to extra loop
suppression or the non-existent gluonic LNV operators) and those being
cancelled out by symmetries (like B-only operators). The remaining
coefficients are indeed renormalised, and the contributions in their equations
are sorted out. In particular, we have found that four insertions of Weinberg
operators can only renormalise the 8-Higgs operator, and they are the only
contribution to 1PI diagrams for this class of operators. On the other hand,
all other operators show at least one term containing the insertion of one
$d_{7}$ operator.

We find surprising the fact that most of the anomalous dimension matrix terms
for operators with more than four Higgs legs (both of dimension-6 and
dimension-8) deviate from the naive dimensional analysis expectation order
$\mathcal{O}(\gamma)\sim 1$. Table A shows that the insertion of a $d_{6}$
operator is the contribution with the highest numerical coefficients (in
absolute value). More specifically, operators of class $\phi^{6}D^{2}$ are
enhanced by factors up beyond $\mathcal{O}(\gamma)\sim 50$ from the insertion
of two Weinberg operators and $\mathcal{O}_{\phi\ell}^{(1,3)}$.

We have discussed that the positivity bounds impose restrictions on
dimension-8 WCs and these restrictions could be translated to LNVs. We have
shown explicitly one such case using the RGE equation of $c_{\phi^{4}}^{(2)}$
operators and its positivity bound, deducing the constraints on the WCs of
$\mathcal{O}_{\ell\phi}^{(5)}$ and $\mathcal{O}_{\ell\phi D}^{(2)}$. We have
indicated that assuming single flavor index, these $d_{5}$ and $d_{7}$ WCs
must have identical signs in any UV completion that does not generate
$\phi^{4}D^{4}$ at tree level. We have inspected these constraints in models
extended with heavy fermions where these LNVs are generated at tree-level, but
the $\phi^{4}D^{4}$ class operators are absent. We have found that these
bounds are respected in Type-I and III seesaw models. We also have shown that
the T-parameter bounds impact the LNV space on account of the RGEs. To
elaborate on this, we have derived the restrictions on the space of Weinberg-
like operators based on T-parameter bound. To quantify the impact, we have
shown in Fig. 3 that these puts restrictions on these Wilson coefficients
complementary to those imposed by neutrino masses.

| $d_{5}$ | $d_{5}^{2}$ | $d_{6}$ | $d_{5}^{3}$ | $d_{5}\times d_{6}$ | $d_{7}$ | $d_{5}^{4}$ | $d_{5}^{2}\times d_{6}$ | $d_{6}^{2}$ | $d_{5}\times d_{7}$ | $d_{8}$
---|---|---|---|---|---|---|---|---|---|---|---
$d_{\leq 4}$ (bosonic) |  |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [51] |  |  |  |  |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [27] |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [28]
$d_{\leq 4}$ (fermionic) |  |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [51] |  |  |  |  |  | X |  | X
$d_{5}$ |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [52, 53, 54] |  |  |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [8] |  ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathbf{\checkmark}}$ [8] |  |  |  |  |
$d_{6}$ (bosonic) |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [44] |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [51, 55, 56] |  |  |  |  | This work |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [27] | This work |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [28]
$d_{6}$ (fermionic) |  |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [44] |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [51, 55, 56, 57] |  |  |  |  | X |  | X | X
$d_{7}$ |  |  |  |  ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathbf{\checkmark}}$ [8] |  ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathbf{\checkmark}}$ [8] |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [42, 58] |  |  |  |  |
$d_{8}$ (bosonic) |  |  |  |  |  |  | This work | This work |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [27] | This work |  ${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$ [28]
$d_{8}$ (fermionic) |  |  |  |  |  |  | X | X | X | X |  ${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathbf{\checkmark}}$ [26]

Table 3: State of the art of the SMEFT renormalisation (adapted from Refs.[28,
27]). The rows show the renormalised operators (categorized by dimensions and
statistics). The columns show the operators contributing to RG running. Blank
entries vanish,
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbf{\checkmark}}$
denotes that the complete contribution is available,
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\mathbf{\checkmark}}$
implies that only (but substantial) partial results is present, and, X
indicates that nothing, or very little, is known. The contribution made in
this paper is marked by This work.

## Acknowledgments

We thank Mikael Chala for suggestions and discussions. We also thank José
Santiago and Pablo Olgoso for help with MatchMakerEFT [59], and Guilherme
Guedes, Anisha and Maria Ramos for comments on the manuscript. This work was
supported in part by SRA (Spain) under Grant No. PID2019-106087GB-C21 /
10.13039/501100011033; by the Junta de Andalucía (Spain) under Grants No. FQM-
101, A-FQM-467-UGR18, and P18-FR-4314 (FEDER). ADC is also supported by the
Spanish MINECO under the FPI programme.

##  Appendix A Tables of operators

Dimension 8
---
$\phi^{8}$ | $\mathcal{O}_{\phi^{8}}$ | $(\phi^{\dagger}\phi)^{4}$ |  |
| $\mathcal{O}_{\phi^{6}}^{(1)}$ | $(\phi^{{\dagger}}\phi)^{2}(D_{\mu}\phi^{{\dagger}}D^{\mu}\phi)$ | $\mathcal{O}_{\phi^{6}}^{(2)}$ | $(\phi^{{\dagger}}\phi)(\phi^{{\dagger}}\sigma^{I}\phi)(D_{\mu}\phi^{{\dagger}}\sigma^{I}D^{\mu}\phi)$
$\phi^{6}D^{2}$ | $\mathcal{O}_{\phi^{6}}^{(3)}$ | $(\phi^{\dagger}\phi)^{2}(\phi^{\dagger}D^{2}\phi+\text{h.c.})$ | $\mathcal{O}_{\phi^{6}}^{(4)}$ | $(\phi^{\dagger}\phi)^{2}D_{\mu}(\phi^{\dagger}\mathrm{i}\overleftrightarrow{D}^{\mu}\phi)$
| $\mathcal{O}_{\phi^{4}}^{(1)}$ | $(D_{\mu}\phi^{{\dagger}}D_{\nu}\phi)(D^{\nu}\phi^{{\dagger}}D^{\mu}\phi)$ | $\mathcal{O}_{\phi^{4}}^{(2)}$ | $(D_{\mu}\phi^{{\dagger}}D_{\nu}\phi)(D^{\mu}\phi^{{\dagger}}D^{\nu}\phi)$
| $\mathcal{O}_{\phi^{4}}^{(3)}$ | $(D^{\mu}\phi^{{\dagger}}D_{\mu}\phi)(D^{\nu}\phi^{{\dagger}}D_{\nu}\phi)$ | $\mathcal{O}_{\phi^{4}}^{(4)}$ | $D_{\mu}\phi^{\dagger}D^{\mu}\phi(\phi^{\dagger}D^{2}\phi+\text{h.c.})$
| $\mathcal{O}_{\phi^{4}}^{(5)}$ | $D_{\mu}\phi^{\dagger}D^{\mu}\phi(\phi^{\dagger}\mathrm{i}D^{2}\phi+\text{h.c.})$ | $\mathcal{O}_{\phi^{4}}^{(6)}$ | $(D_{\mu}\phi^{\dagger}\phi)(D^{2}\phi^{\dagger}D_{\mu}\phi)+\text{h.c.}$
| $\mathcal{O}_{\phi^{4}}^{(7)}$ | $(D_{\mu}\phi^{\dagger}\phi)(D^{2}\phi^{\dagger}\mathrm{i}D_{\mu}\phi)+\text{h.c.}$ | $\mathcal{O}_{\phi^{4}}^{(8)}$ | $(D^{2}\phi^{\dagger}\phi)(D^{2}\phi^{\dagger}\phi)+\text{h.c.}$
| $\mathcal{O}_{\phi^{4}}^{(9)}$ | $(D^{2}\phi^{\dagger}\phi)(\mathrm{i}D^{2}\phi^{\dagger}\phi)+\text{h.c.}$ | $\mathcal{O}_{\phi^{4}}^{(10)}$ | $(D^{2}\phi^{\dagger}D^{2}\phi)(\phi^{\dagger}\phi)$
| $\mathcal{O}_{\phi^{4}}^{(11)}$ | $(\phi^{\dagger}D^{2}\phi)(D^{2}\phi^{\dagger}\phi)$ | $\mathcal{O}_{\phi^{4}}^{(12)}$ | $(D_{\mu}\phi^{\dagger}\phi)(D^{\mu}\phi^{\dagger}D^{2}\phi)+\text{h.c.}$
$\phi^{4}$ | $\mathcal{O}_{\phi^{4}}^{(13)}$ | $(D_{\mu}\phi^{\dagger}\phi)(D^{\mu}\phi^{\dagger}\mathrm{i}D^{2}\phi)+\text{h.c.}$ |  |
| $O_{G^{2}\phi^{4}}^{(1)}$ | $(\phi^{\dagger}\phi)^{2}G_{\mu\nu}^{A}G^{A\mu\nu}$ | $O_{G^{2}\phi^{4}}^{(2)}$ | $(\phi^{\dagger}\phi)^{2}\widetilde{G}_{\mu\nu}^{A}G^{A\mu\nu}$
| $\mathcal{O}_{W^{2}\phi^{4}}^{(1)}$ | $(\phi^{\dagger}\phi)^{2}W^{I}_{\mu\nu}W^{I\mu\nu}$ | $\mathcal{O}_{W^{2}\phi^{4}}^{(2)}$ | $(\phi^{\dagger}\phi)^{2}\widetilde{W}^{I}_{\mu\nu}W^{I\mu\nu}$
| $\mathcal{O}_{W^{2}\phi^{4}}^{(3)}$ | $(\phi^{\dagger}\sigma^{I}\phi)(\phi^{\dagger}\sigma^{J}\phi)W^{I}_{\mu\nu}W^{J\mu\nu}$ | $\mathcal{O}_{W^{2}\phi^{4}}^{(4)}$ | $(\phi^{\dagger}\sigma^{I}\phi)(\phi^{\dagger}\sigma^{J}\phi)\widetilde{W}^{I}_{\mu\nu}W^{J\mu\nu}$
| $\mathcal{O}_{WB\phi^{4}}^{(1)}$ | $(\phi^{\dagger}\phi)(\phi^{\dagger}\sigma^{I}\phi)W^{I}_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{WB\phi^{4}}^{(2)}$ | $(\phi^{\dagger}\phi)(\phi^{\dagger}\sigma^{I}\phi)\widetilde{W}^{I}_{\mu\nu}B^{\mu\nu}$
$X^{2}\phi^{4}$ | $\mathcal{O}_{B^{2}\phi^{4}}^{(1)}$ | $(\phi^{\dagger}\phi)^{2}B_{\mu\nu}B^{\mu\nu}$ | $\mathcal{O}_{B^{2}\phi^{4}}^{(2)}$ | $(\phi^{\dagger}\phi)^{2}\widetilde{B}_{\mu\nu}B^{\mu\nu}$
| $\mathcal{O}_{W\phi^{4}D^{2}}^{(1)}$ | $\text{i}(\phi^{{\dagger}}\phi)(D^{\mu}\phi^{{\dagger}}\sigma^{I}D^{\nu}\phi)W_{\mu\nu}^{I}$ | $\mathcal{O}_{W\phi^{4}D^{2}}^{(2)}$ | $\text{i}(\phi^{{\dagger}}\phi)(D^{\mu}\phi^{{\dagger}}\sigma^{I}D^{\nu}\phi)\widetilde{W}_{\mu\nu}^{I}$
| $\mathcal{O}_{W\phi^{4}D^{2}}^{(3)}$ | $\text{i}\epsilon^{IJK}(\phi^{{\dagger}}\sigma^{I}\phi)(D^{\mu}\phi^{{\dagger}}\sigma^{J}D^{\nu}\phi)W_{\mu\nu}^{K}$ | $\mathcal{O}_{W\phi^{4}D^{2}}^{(4)}$ | $\text{i}\epsilon^{IJK}(\phi^{{\dagger}}\sigma^{I}\phi)(D^{\mu}\phi^{{\dagger}}\sigma^{J}D^{\nu}\phi)\widetilde{W}_{\mu\nu}^{K}$
| $\mathcal{O}_{W\phi^{4}D^{2}}^{(5)}$ | $(\phi^{\dagger}\phi)D_{\nu}W^{I\mu\nu}(D_{\mu}\phi^{\dagger}\sigma^{I}\phi+\text{h.c.})$ | $\mathcal{O}_{W\phi^{4}D^{2}}^{(6)}$ | $(\phi^{\dagger}\phi)D_{\nu}W^{I\mu\nu}(D_{\mu}\phi^{\dagger}\text{i}\sigma^{I}\phi+\text{h.c.})$
| $\mathcal{O}_{W\phi^{4}D^{2}}^{(7)}$ | $\epsilon^{IJK}(D_{\mu}\phi^{\dagger}\sigma^{I}\phi)(\phi^{\dagger}\sigma^{J}D_{\nu}\phi)W^{K\mu\nu}$ |  |
| $\mathcal{O}_{B\phi^{4}D^{2}}^{(1)}$ | $\text{i}(\phi^{{\dagger}}\phi)(D^{\mu}\phi^{{\dagger}}D^{\nu}\phi)B_{\mu\nu}$ | $\mathcal{O}_{B\phi^{4}D^{2}}^{(2)}$ | $\text{i}(\phi^{{\dagger}}\phi)(D^{\mu}\phi^{{\dagger}}D^{\nu}\phi)\widetilde{B}_{\mu\nu}$
$X\phi^{4}D^{2}$ | $\mathcal{O}_{B\phi^{4}D^{2}}^{(3)}$ | $(\phi^{{\dagger}}\phi)D_{\nu}B^{\mu\nu}(D_{\mu}\phi^{\dagger}\text{i}\phi+\text{h.c.})$ |  |

Table 4: Dimension-8 bosonic operators with four or more Higgs fields in the
Green’s basis of Ref. [41]. Operators in gray are redundant.

Dimension 7
---
$\psi^{2}\phi^{4}$ | $\mathcal{O}_{\ell\phi}^{(7)}$ | $\epsilon_{ij}\epsilon_{mn}(\ell^{i}C\ell^{m})(\phi^{j}\phi^{n})(\phi^{\dagger}\phi)$ |  |
$\psi^{2}\phi^{2}D^{2}$ | $\mathcal{O}_{\ell\phi D}^{(1)}$ | $\epsilon_{ij}\epsilon_{mn}\ell^{i}C(D^{\mu}\ell^{j})\phi^{m}(D_{\mu}\phi^{n})$ | $\mathcal{O}_{\ell\phi D}^{(2)}$ | $\epsilon_{im}\epsilon_{jn}\ell^{i}C(D^{\mu}\ell^{j})\phi^{m}(D_{\mu}\phi^{n})$
$\psi^{2}\phi^{3}D$ | $\mathcal{O}_{\ell\phi De}$ | $\epsilon_{ij}\epsilon_{mn}(\ell^{i}C\gamma_{\mu}e)\phi^{j}\phi^{m}D^{\mu}\phi^{n}$ |  |
$\psi^{2}\phi^{2}X$ | $\mathcal{O}_{\ell\phi B}$ | $\epsilon_{ij}\epsilon_{mn}\ell^{i}C(\sigma_{\mu\nu}\ell^{m})\phi^{j}\phi^{n}B^{\mu\nu}$ | $\mathcal{O}_{\ell\phi W}$ | $\epsilon_{ij}(\tau^{I}\epsilon)_{mn}\ell^{i}C(\sigma_{\mu\nu}\ell^{m})\phi^{j}\phi^{n}W^{I\mu\nu}$
Dimension 6
$\phi^{6}$ | $\mathcal{O}_{\phi}$ | $(\phi^{\dagger}\phi)^{3}$ |  |
| $\mathcal{O}_{\phi\square}$ | $(\phi^{\dagger}\phi)\Square(\phi^{\dagger}\phi)$ | $\mathcal{O}_{\phi D}$ | $(\phi^{\dagger}D^{\mu}\phi)^{\dagger}(\phi^{\dagger}D_{\mu}\phi)$
$\phi^{4}D^{2}$ | $\mathcal{O}_{\phi D}^{\prime}$ | $(\phi^{\dagger}\phi)(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)$ | $\mathcal{O}_{\phi D}^{\prime\prime}$ | $\text{i}(\phi^{\dagger}\phi)D_{\mu}(\phi^{\dagger}D^{\mu}\phi-D^{\mu}\phi^{\dagger}\phi)$
| $\mathcal{O}_{u\phi}$ | $(\phi^{\dagger}\phi)\overline{q}\widetilde{\phi}u$ | $\mathcal{O}_{d\phi}$ | $(\phi^{\dagger}\phi)\overline{q}\phi d$
$\psi^{2}\phi^{3}$ | $\mathcal{O}_{e\phi}$ | $(\phi^{\dagger}\phi)\overline{\ell}\phi e$ |  |
| $\mathcal{O}^{(1)}_{Hq}$ | $\mathrm{i}(\overline{q}\gamma^{\mu}q)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$ | $\mathcal{O}^{(3)}_{Hq}$ | $\mathrm{i}(\overline{q}\sigma^{I}\gamma^{\mu}q)(\phi^{\dagger}\sigma^{I}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\sigma^{I}\phi)$
| $\mathcal{O}_{Hu}$ | $\mathrm{i}(\overline{u}\gamma^{\mu}u)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$ | $\mathcal{O}_{Hd}$ | $\mathrm{i}(\overline{d}\gamma^{\mu}d)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$
| $\mathcal{O}^{(1)}_{H\ell}$ | $\mathrm{i}(\overline{\ell}\gamma^{\mu}\ell)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$ | $\mathcal{O}^{(3)}_{H\ell}$ | $\mathrm{i}(\overline{\ell}\sigma^{I}\gamma^{\mu}\ell)(\phi^{\dagger}\sigma^{I}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\sigma^{I}\phi)$
$\psi^{2}\phi^{2}D$ | $\mathcal{O}_{He}$ | $\mathrm{i}(\overline{e}\gamma^{\mu}e)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$ | $\mathcal{O}_{Hud}$ | $\mathrm{i}(\overline{u}\gamma^{\mu}d)(\phi^{\dagger}D_{\mu}\phi-D_{\mu}\phi^{\dagger}\phi)$
Dimension 5
$\psi^{2}\phi^{2}$ | $\mathcal{O}_{\ell\phi}^{(5)}$ | $\epsilon_{ij}\epsilon_{mn}(\ell^{i}C\ell^{m})(\phi^{j}\phi^{n})$

Table 5: Dimension-6 and -7 from bases in Refs. [40] and [39], needed for the
renormalisation of dimension-8 bosonic operators.

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