Re-examining NR-EFT Upto Dimension Six Manimala Mitra1 2Sanjoy Mandal3yRojalin Padhan1 2 4zAgnivo Sarkar1 2xand Michael Spannowsky5 1Institute of Physics Sachivalaya Marg Bhubaneswar 751005 India

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Re-examining NR-EFT Upto Dimension Six

Manimala Mitra,1, 2, ∗Sanjoy Mandal,3, †Rojalin Padhan,1, 2, 4, ‡Agnivo Sarkar,1, 2, §and Michael Spannowsky5, ¶

1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India

2Homi Bhabha National Institute, BARC Training School Complex, Anushakti Nagar, Mumbai 400094, India

3Korea Institute for Advanced Study, Seoul 02455, Korea

4Pittsburgh Particle Physics, Astrophysics, and Cosmology Center, Department of Physics and Astronomy,

University of Pittsburgh, Pittsburgh, USA

5Institute for Particle Physics Phenomenology, Department of Physics, Durham University

South Road, Durham DH1 3LE, United Kingdom

The gauge singlet right-handed neutrinos (RHNs) are essential ﬁelds in several neutrino mass

models that explain the observed eV scale neutrino mass. We assume RHN ﬁeld to be present in the

vicinity of the electroweak scale and all the other possible beyond the standard model (BSM) ﬁelds

arise at high energy scale ≥Λ. In this scenario, the BSM physics can be described using eﬀective ﬁeld

theory (EFT) where the set of canonical degrees of freedoms consists of both RHN and SM ﬁelds.

EFT of this kind is usually dubbed as NR-EFT. We systematically construct relevant operators

that can arise at dimension ﬁve and six while respecting underlying symmetry. To quantify the

phenomenological implication of these EFT operators we calculate diﬀerent couplings that involve

RHN ﬁelds. We discuss the constraints on these EFT operators coming from diﬀerent energy and

precision frontier experiments. For pp,e−pand e+e−colliders, we identify various channels which

crucially depends on these operators. We analytically evaluate the decay widths of RHN considering

all relevant operators and highlight the diﬀerences that arise because of the EFT framework. Based

upon the signal cross-section we propose diﬀerent multi-lepton channels to search for the RHN at

14 TeV LHC as well as future particle colliders.

1. INTRODUCTION

The tremendous achievement of the Standard Model (SM) is that it can make precise numerical predictions about

the particle dynamics up to the TeV scale. The Higgs boson’s discovery [1, 2] at the Large Hadron Collider (LHC) as

well as precision frontier experiments favour the theoretical claims of this model with signiﬁcant precision. Despite

these experimental success, there are many compelling reasons correspond to non-zero neutrino mass, dark matter or

the natural explanation behind the electroweak symmetry breaking etc. motivate us to construct Beyond Standard

Model (BSM) theories which can satisfactorily explain these questions. These BSM theories typically contain new

degrees of freedom (d.o.f ) which interact with the SM particles. Diﬀerent experimental collaborations have extensively

looked for these BSM particles decaying into various SM ﬁnal states. The results obtained from these searches so

far fail to provide any conclusive evidence in support of their existence or their corresponding properties. One of the

plausible explanations behind these null results is that these BSM states are situated at a very large energy scale Λ

and the centre of mass energy of the present day colliders is not suﬃcient enough to produce them on-shell. However

the indirect eﬀects of these particles can be detected while analysing diﬀerent low-energy observables [3]. In view

of this, one can consider the eﬀective ﬁeld theory (EFT) [4, 5] approach which can serve as an eﬃcient pathway to

parametrise these indirect eﬀects that can help us uncover the nature of BSM.

The construction of any EFT [6, 7] typically requires two ingredients, the canonical degrees of freedom (d.o.f )

that are present in low energy theory and the symmetries which manifestly dictate the interactions between these

fundamental d.o.f. The Lagrangian corresponds to the EFT framework [8] is sum of both the d= 4 renormalisable

part as well as diﬀerent higher dimensional operators which are allowed by the symmetry. We assume at the scale

Λ, their exists a gauge theory which contains extra massive d.o.f. At this scale these ﬁelds get decouple from the

low-energy theory. The eﬀects of these heavy states can be reinstated in forms of a tower of eﬀective operators at each

order of mass dimensions n > 4. These higher dimensional operators {On}1are built upon canonical d.o.f of low

energy theory while respecting space-time as well as the gauge and discrete symmetries. The decoupling theorem [9, 10]

guarantees that all measurable observables corresponding to the heavy scale physics are suppressed by inverse powers

of cut-oﬀ scale Λ. As a corollary of the decoupling theorem one can establish the hierarchy between the operators

∗manimala@iopb.res.in

†smandal@kias.re.kr

‡rojalin.p@iopb.res.in

§agnivo.sarkar@iopb.res.in

¶michael.spannowsky@durham.ac.uk

1The nstands for the mass dimension of these operators.

arXiv:2210.12404v2 [hep-ph] 29 Oct 2022

that arise at each dimension. As a consequence, the measurable eﬀects of the operators at dimension nin general

dominant over the operators arise at dimension n+1. One can optimally use this framework to investigate the physics

associated with neutrinos and establish their connection with the SM physics.

The absence of RHNs (N) in the SM ﬁeld content, forbids us to generate neutrino mass similar to other SM

fermions. However, various neutrino oscillations experimental measurements [11–15] strongly suggest non zero masses

for neutrinos thus encourages us to modify the existing SM. The simplest way to encounter this issue is to add

RHNs to the SM particle contents and write down a Yukawa interaction for neutrinos similar to other SM charged

fermions. As these RHN ﬁelds are charge neutral and singlet under the SM gauge group SU(3)c×SU(2)L×U(1)Y,

one can include a Lepton-number violating Majorana type mass term MNNc

RNRin the Lagrangian in addition to

the previously mentioned Yukawa interaction. The smallness of the neutrino mass can therefore be explained as the

hierarchy between the electroweak scale vand the RHN mass scale MNwhich can be expressed as Mν∼y2

νv2

MN. Here,

yνstands for Yukawa coupling correspond to neutrinos. If we assume the value of yνto be O(1), one can see that the

requirement for tiny neutrino mass set the value of MNin the vicinity of Grand Uniﬁcation regime (roughly around

1014 ∼1015 GeV). This simplistic set up for neutrino mass is in general known as Type-I Seesaw mechanism [16–19].

The interaction strength between these heavy neutrinos and the SM particles is controlled by the active sterile mixing

parameter θwhich is deﬁned as θ∝yνv

MN. The above relation implies a small value of θand leads to a small production

cross-section for the RHNs at diﬀerent collider experiments.

The major disadvantage of the Type-I set up is that the physics associated with the RHN ﬁelds become relevant at

around GUT scale which the current experimental facilities fail to probe. One can alter this situation while assuming

that at least one of these RHN ﬁelds is within the regime of electroweak scale [20–22] while satisfying the existing

experimental constraints. In this context one can describe the dynamics involving RHN using EFT. The EFT of this

kind is denoted as NR-EFT.

There are many works which encompass diﬀerent aspects of NR-EFT. The Ref. [23–26] and Ref. [27] presents

the non-redundant operator basis upto dimension seven and dimension nine of NR-EFT respectively. The Ref. [28–

31] discuss the collider phenomenology of the dimension ﬁve NR-EFT at future Higgs factories as well as LHC.

Other studies [23, 32–35] also looked into various subset of these higher dimensional operators and presented their

phenomenological implication at LHC. If the total decay width of the light RHN is small then it can give rise to

interesting displaced decay signatures and detailed study regarding this can be found in Ref. [36–39]. The Ref. [40–42]

focused on the interesting production modes which is invoked by the diﬀerent four fermi operators that one construct

at dimension six. The study assume relevant decay modes for the Nﬁeld to be N→νγ and N→3f(where f

is SM fermions). The Ref. [43] discuss the theoretical aspects of the dimension 6 operators that involve the Higgs

doublet and discuss their sensitivity under various Higgs mediated processes. In addition to that, Ref. [44, 45] study

the sensitivity of diﬀerent dimension six operators at LHeC and lepton colliders.

In this work we present the complete phenomenological description of the NR-EFT upto dimension six. In section 2

we begin with the general set up and systemically construct diﬀerent dimension ﬁve (see sub-section 2.1) as well as

dimension six (see sub-section 2.2) operators along with highlighting their physics aspects. In section 3, we evaluate

the constraints on diﬀerent operators coming from precision frontier as well as direct search experiments. In section 4,

we calculate the cross section for RHN production at pp,e−pand e+e−colliders. Depending on the RHN mass,

the Nﬁeld can decay either to two body or to three body decay modes respectively. In section 5, we present the

detailed analytic calculations correspond to each of these decay modes and evaluate the branching ratios for diﬀerent

benchmark scenarios. We also present expected number of signal events with multi-lepton ﬁnal state for above

mentioned colliders in section. 6. We summarise our ﬁndings along with few concluding remarks in section 7.

2. GENERAL SET UP

We begin with a phenomenological Lagrangian which can be expressed as

L≡LSM +¯

NR

∂NR−¯

L`Yν˜

HNR−1

2˜

MN¯

RNR+X

n>4

Λn−4+h.c. (2.1)

where ˜

H=iσ2H∗,NC

R=C¯

Rwith charge conjugation matrix C=iγ2γ0. The term ˜

MNstands for the Majorana

bare mass term which is a N ×N matrix in the ﬂavour space. L`is the SM lepton doublet and Yνis the Dirac-type

Yukawa coupling. The terms ¯

L`Yν˜

HNRand 1

2˜

MN¯

RNRcontributes to the neutrino mass matrix. The Onare the

higher dimensional operators, which one can build at each dimension. The eﬀect of these operators are suppressed by

cut-oﬀ scale Λ with appropriate power.

2.1. NR-EFT Operators at Dimension Five

With this general set-up in mind, one can write down three possible NR-EFT operators at dimension ﬁve. In

Table.I, we present the explicit form of these operators where α(5)

i(i= 1 to 3 ) represent the Wilson coeﬃcients

correspond to each of these operators. Considering the space time transformation rules, one can realise that the α(5)

O(5)

α(5)

ΛLc˜

H†˜

HL

O(5)

α(5)

ΛNR

cNRH†H

O(5)

α(5)

ΛNR

cσµν NRBµν

TABLE I. All Possible NR-EFT operators that appear at dimension ﬁve. The σµν is deﬁned as, σµν =i

2[γµ, γν] and Bµν is

the ﬁeld strength tensor corresponds to U(1)Ygauge group. Λ is the cut-oﬀ scale of underlying NR-EFT.

and α(5)

2are symmetric matrices in ﬂavour space. In contrast to that, α(5)

3is an antisymmetric matrix which arises

if we only consider more than one NRﬁelds. The O(5)

1which is famously known as the Weinberg operator [46]

primarily contributes to active neutrino masses. This is the only operator one can construct in this dimension solely

using SM ﬁelds. The renormalisable realisation of this operator can be found in Ref. [47–49] and its phenomenological

implications have been studied in Ref. [50, 51]. On the other hand, operator O(5)

2provides additional contributions

to the Majorana mass term which is mentioned in Eq. 2.2. However the operator O(5)

3does not play any role in the

neutrino mass matrix but the presence of Bµν in that term brings out non trivial vertices between neutrinos and

SM neutral vector boson ﬁelds. Assuming the full theory is a gauge theory one may predict that out of these three

operators, O(5)

1and O(5)

2may be generated in tree level but the O(5)

3would only appear via loop mediated processes.

As a consequence, one can estimate a further 1

16π2suppression to the α(5)

3coeﬃcient [52]. For a detailed discussion

on this aspect the interested reader may follow Ref. [53].

•Neutrino Mass In Dimension Five

We will now deﬁne the neutrino mass matrix while considering all the relevant terms upto dimension ﬁve. In the basis

{νL, Nc

R}, the neutrino mass matrix will take the following form

M(5)

νN =





α(5)

1v2

Yνv

√2

νv

√2˜

MN+α(5)

2v2

Λ



(2.2)

In the seesaw approximation (when ν−Nblocks are smaller than the ones in the N−None), this leads to the

following light and heavy neutrino mass matrix

m(5)

light ≈α(5)

1v2

Λ−YT

νM−1

Nv2Yν

2,(2.3)

m(5)

heavy ≈MN=˜

MN+α(5)

2v2

Λ.(2.4)

The mass matrix in Eq. 2.2 can be diagonalized by a unitary matrix as

VTM(5)

νN V= (M(5)

νN )diag.(2.5)

Following the standard procedure of two step diagonalization Vcan be expressed as

V=UWwith UTM(5)

νN U= m(5)

light 0

0m(5)

heavy!(2.6)

Hence, Uis the matrix which brings the neutrino mass matrix in the block diagonalized form and further W=

Diag(UPMNS, κ) diagonalizes the mass matrices in the light and heavy sector. One can approximately write the

matrix Vas follows

V=UW≈1 + O(M−2

N)θ

−θT1 + O(M−2

N)UPMNS 0

0κ≈UPMNS θ

−θTκ,(2.7)

where θ=M−1

Yνv

√2is the mixing angle between the active and sterile neutrinos, UPMNS is the PMNS matrix and κ

is O(1) (For details see Ref. [54]). Following is the mixing relations between the gauge and mass eigenstates

νL'UPMNSνL,m +θNc

R,m,(2.8)

R' −θTνL,m +κNc

R,m,

where the subscript “m” signiﬁes the mass eigenstate.

•Interesting Facets of the Dimension Five Operators

In Eq. 2.8, we show the relation between ﬂavour and mass eigenstates between light (active) and heavy (sterile)

neutrinos. In the subsequent discussion, we denote the Majorana mass eigenstate of RHN ﬁelds as N=NR,m +

R,m, while we use similar notation for light neutrino mass basis, ν=νL,m +νc

L,m. With these deﬁnitions we now

present various three point vertices that involve neutrino ﬁelds which are coming from renormalizable Lagrangian and

dimension ﬁve operators. The details of the calculations have been included in Appendix B. In Table.II, we illustrate

the explicit form of all these couplings. One can notice that the coupling between the Wµboson and neutrinos does

not get any additional contributions from the dimension ﬁve operators. However, the situation alters in case of Higgs

as well as neutral gauge boson operators.

Couplings Explicit Form Operator

CWµ

`ν

gγµU

√2PL+ h.c. RT

CWµ

gγµθ

√2PL+ h.c. RT

νν Yν

√2U†θ†PR+α(5)

ΛUTUPL+α(5)

Λθ∗θ†PR+ h.c. RT, O(5)

1,O(5)

NN −Yν

√2θ†κ∗PR+α(5)

Λθ†θPL+α(5)

Λκ†κ∗PR+ h.c. RT, O(5)

1,O(5)

νN+Nν {−Yν

√2U†κ∗PR+α(5)

ΛU†θPL−α(5)

Λθ∗κ∗PR}

+{Yν

√2θ†θ†PR+α(5)

Λθ†UPL−α(5)

Λκ†θ†PR}+ h.c. RT, O(5)

1,O(5)

CZµ

νν

gγµ

2cwU†UPL−2iα(5)

3sw

Λθ∗θpνσµν PR+ h.c. RT, O(5)

CZµ

gγµ

2cwθ†θPL−2iα(5)

3sw

Λκ†κ∗pνσµν PR+ h.c. RT, O(5)

CZµ

νN+Nν {gγµ

2cwU†θPL+ 2iα(5)

3sw

Λθ∗κ∗pνσµν PR}

+{gγµ

2cwUθ†PL+ 2iα(5)

3sw

Λκ†θ†pνσµν PR}+ h.c. RT, O(5)

CAµ

νν 2iα(5)

3cw

Λθ∗θ†pνσµν PR+ h.c. O(5)

CAµ

NN 2iα(5)

3cw

Λκ†κ∗pνσµν PR+ h.c. O(5)

CAµ

νN+Nν {−2iα(5)

3cw

Λθ∗κ∗pνσµν PR} −{2iα(5)

3cw

Λκ†θ†pνσµν PR}+ h.c. O(5)

TABLE II. Coupling from the three-point vertices that arise after taking into account both the dimension four and dimension

ﬁve terms of the Lagrangian. Here Usigniﬁes UPMNS matrix. The abbreviation “RT” stands for renormalisable term which

includes charge current, neutral current as well as Yukawa term. The chirality projection matrix is denoted by PLand PR.

The momentum factor pνin diﬀerent vertices arise from ﬁeld strength tensor Bµν after transforming it into momentum space.

•The tree-level vertices that involve Higgs ﬁeld do get modiﬁed due to the presence of O(5)

1,O(5)

2operators. In

view of Eq. 2.4, one can see that the operator O(5)

1regulates the SM neutrino masses. The smallness of these

mass values forces us to choose a tiny magnitude for α(5)

Λ, which is below the order of O10−11GeV for Λ to

be in the order TeV. This is why, the eﬀects coming from this operator can not be studied in the present day

experimental set up. Due to this, for all our analysis we will set α(5)

1to be zero.

•In contrast to that, a similar conclusion can not be made for α(5)

Λcoeﬃcient. Hence, one should critically analyse

it’s role on a case by case basis.

•The operator O(5)

3changes the couplings that involve both massless and massive vector boson ﬁelds. However

as we have mentioned before, the structure of this operator contains two important aspects. First, the α(5)

3is

an anti-symmetric matrix in the ﬂavour space and can only exist if we consider more than one ﬂavour of NR

ﬁelds within the EFT framework. In addition to that, from a full theory point of view the vertices coming from

this operators can not possibly be realised in tree level graphs. Hence, the eﬀects come from this operator must

be further suppressed by the loop factor 1

16π2.

•The compelling facet of the operator O(5)

3is to invoke a non-trivial coupling between the photon ﬁeld and

neutrinos which are not present in the SM counterpart. The presence of Bµν tensor in the O(3)

5operator

introduce interaction term between the RHN ﬁelds and hyper-charge gauge boson Bµ. After the symmetry

breaking the Bµﬁeld can be written as the linear combination of Zboson and photon (Bµ=−swZµ+cwAµ).

As a consequence of the ﬁeld redeﬁnition, NRﬁelds would couple to photon. These couplings would have a

direct impact to neutrino magnetic moment [53]. Using XENON data [55] one can determine the size of the

associated Wilson coeﬃcient α(5)

Λ. Here we conclude our discussion on dimension ﬁve operators and in the

subsequent section, we discuss d= 6 operators.

2.2. NR-EFT Operators at Dimension Six

In the last section we have presented various aspects of dimension ﬁve operators. We will now turn our attention

to the details of the dimension six operators. In Table. III, we enlist all possible operators in systematic manner. For

a methodical construction of these operators, one may read through Ref. [26].

•Neutrino Mass In Dimension Six

Before engaging ourselves into an extensive discussion on these operators, we like to point out possible modiﬁcation

happens in the neutrino mass matrix when one consider dimension six operators. The operator that falls under the

class of ψ2H3, where ψ2represents two fermionic ﬁelds, contributes towards the neutrino mass matrix as this operator

would give additional contribution towards the oﬀ-diagonal Dirac elements of the matrix mentioned in Eq. 2.2. The

updated form of this matrix can be illustrated in the following fashion -

M(6)

νN =





α(5)

1v2

Yνv

√2+αLNH v3

2√2Λ2

νv

√2+αT

LNH v3

2√2Λ2˜

MN+α(5)

2v2

Λ



.(2.9)

Our next task is to obtain the correct form of eigenvalues and eigenvectors corresponds to the light and heavy neutrinos

respectively. To do so, we would consider the following re-deﬁnition of the oﬀ-diagonal element of the above matrix.

Yν=Yν+αLNH v2

2Λ2(2.10)

We use this parametrisation, to write down the light neutrino mass. To evaluate the eigenvalues of the above matrix

we choose the limit MNαLNH v3

Λ2,α(5)

1v2

Λ. In this limit, the light and heavy mass eigenvalue will take the following

matrix form

m(6)

light ≈α(5)

1v2

Λ−˜

ν(˜

M−1

N)v2˜

Yν

m(6)

heavy ≈MN.(2.11)

Looking at the above form of the neutrino mass matrices one can appreciate the rational behind the parametrisation

mentioned in Eq. 2.10. The inclusion of the dimension six contribution does not alter the form of the mass eigenvalues

as compare to Eq. 2.4. The mass matrix in the dimension six set up can also be diagonalised using the prescription

discussed in the last section. To do so we need to re-deﬁne the mixing angle between active and sterile neutrino. The

matrix Vof Eq. 2.7 will take the following form

V≈UPMNS ˜

−˜

θTκ,(2.12)

where ˜

θis -

θ=θ(5) +θ(6) =M−1

Yνv

√2,and θ(5) =M−1

Yνv

√2, θ(6) =M−1

αLNH v3

2√2Λ2

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Re-examiningNR-EFTUptoDimensionSixManimalaMitra,1,2,SanjoyMandal,3,yRojalinPadhan,1,2,4,zAgnivoSarkar,1,2,xandMichaelSpannowsky5,{1InstituteofPhysics,SachivalayaMarg,Bhubaneswar751005,India2HomiBhabhaNationalInstitute,BARCTrainingSchoolComplex,AnushaktiNagar,Mumbai400094,India3KoreaInstituteforAdva...

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Re-examining NR-EFT Upto Dimension Six Manimala Mitra1 2Sanjoy Mandal3yRojalin Padhan1 2 4zAgnivo Sarkar1 2xand Michael Spannowsky5 1Institute of Physics Sachivalaya Marg Bhubaneswar 751005 India

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