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JHEP06(2019)055

Published for SISSA by Springer

Received: May 22, 2019

Accepted: June 7, 2019

Published: June 13, 2019

On the determination of leptonic CP violation and

neutrino mass ordering in presence of non-standard

interactions: present status

Ivan Esteban,

M.C. Gonzalez-Garcia

b,a,c

and Michele Maltoni

Departament de Fis´ıca Qu`antica i Astrof´ısica and Institut de Ciencies del Cosmos,

Universitat de Barcelona,

Diagonal 647, E-08028 Barcelona, Spain

C.N. Yang Institute for Theoretical Physics,

State University of New York at Stony Brook,

Stony Brook, NY 11794-3840, U.S.A.

Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),

Pg. Lluis Companys 23, 08010 Barcelona, Spain

Instituto de F´ısica Te´orica UAM/CSIC, Universidad Aut´onoma de Madrid,

Calle de Nicol´as Cabrera 13–15, Cantoblanco, E-28049 Madrid, Spain

E-mail: ivan.esteban@fqa.ub.edu,

maria.gonzalez-garcia@stonybrook.edu, michele.maltoni@csic.es

Abstract: We perform a global analysis of neutrino data in the framework of three massive

neutrinos with non-standard neutrino interactions which aﬀect their evolution in the matter

background. We focus on the eﬀect of NSI in the present observables sensitive to leptonic

CP violation and to the mass ordering. We consider complex neutral current neutrino

interactions with quarks whose lepton-ﬂavor structure is independent of the quark type. We

quantify the status of the “hints” for CP violation, the mass-ordering and non-maximality

of θ

in these scenarios. We also present a parametrization-invariant formalism for leptonic

CP violation in presence of a generalized matter potential induced by NSI.

Keywords: Neutrino Physics, Solar and Atmospheric Neutrinos

ArXiv ePrint: 1905.05203

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP06(2019)055

JHEP06(2019)055

Contents

1 Introduction 1

2 Formalism 3

2.1 NSI-mass-ordering degeneracy 4

2.2 New sources of CP violation 5

3 Analysis framework 8

4 Results 10

5 Summary 16

A Invariants for leptonic CP violation with NSI 16

1 Introduction

Experiments measuring the ﬂavor composition of neutrinos produced in the Sun, in the

Earth’s atmosphere, in nuclear reactors and in particle accelerators have established that

lepton ﬂavor is not conserved in neutrino propagation, but it oscillates with a wavelength

which depends on distance and energy. This demonstrates beyond doubt that neutrinos

are massive and that the mass states are non-trivial admixtures of ﬂavor states [1, 3], see

ref. [4] for an overview.

The most recent global analysis of the neutrino oscillation data in ref. [5] (see also [6, 7])

in the context of 3ν-mixing provides us with the determination of the three leptonic mixing

angles and the two mass diﬀerences with a 1σ precision of 3% for θ

, 3% for θ

, 9%

for θ

, 5% for ∆m

and 2.5% for |∆m

|. Questions still open in the analysis are the

maximality and octant of θ

, the ordering of the mass eigenstates, and the status of

the leptonic CP violating phase δ

. Some hints in this respect are emerging — with a

special role played by the long-baseline (LBL) accelerator experiments T2K [8, 9], and

NOvA [10, 11] — but without a consolidated statistical signiﬁcance yet. The latest results

show a preference of the normal ordering (NO) at the 2–3σ level, and a best ﬁt for the

complex phase at δ

= 215

◦

close to maximal CP violation. The clariﬁcation of these

unknowns is the main focus of the running LBL experiments and its precise determination

is at the center of the physics program of the upcoming LBL facilities, in particular the

Deep Underground Neutrino Experiment (DUNE) [12] and the Tokay to HyperKamiokande

(T2HK) experiment [13].

Under the assumption that the Standard Model (SM) is the low energy eﬀective model

of a complete high energy theory, neutrino masses emerge naturally as the ﬁrst observable

– 1 –

JHEP06(2019)055

consequence in the form of the Weinberg operator [14], the only dimension ﬁve operator

that can be built within the SM particle content. In this framework the next operators

with observable consequences at low energies appear at dimension six. They include four-

fermion terms leading to Non-Standard Interactions (NSI) [15–17] between neutrinos and

matter (for a recent review see [18]).

Neutral Current NSI can modify the forward-coherent scattering (i.e., at zero momen-

tum transfer) of neutrinos as they propagate through matter via the so-called Mikheev-

Smirnov-Wolfenstein (MSW) mechanism [15, 19]. Consequently their eﬀect can be signiﬁ-

cantly enhanced in oscillation experiments where neutrinos travel large regions of matter.

Indeed, the global analysis of data from oscillation experiments in the framework of mass

induced oscillations in presence of NSI currently provides some of the strongest constraints

on the size of the NSI aﬀecting neutrino propagation [20–22].

In the presence of such NSI, however, the task of exploring leptonic CP violation in

LBL experiments becomes enriched (to the point of confusion) by the possible coexistence

of new sources of CP violation [23]. Furthermore, the determination of the mass ordering

is jeopardized by the presence of an intrinsic degeneracy in the relevant oscillation prob-

abilities due to a new symmetry of the Hamiltonian describing the neutrino evolution in

the modiﬁed matter potential [20, 21, 24, 25]. This has resulted in an intense phenomeno-

logical activity to quantify these issues and to devise strategies to clarify them, ﬁrst in

proposed facilities like the Neutrino Factory [26–28] and most recently in the context of

the upcoming experiments [29–51].

Very interestingly, it has been argued that NSIs can already play a role in the signiﬁ-

cance of the “hints” mentioned above [52, 53]. In particular in ref. [52] it was pointed out

the discomforting possibility of confusing CP conserving NSI with a non-zero value of δ

in the analysis of ν

and ¯ν

appearance results at T2K and NOvA. Clearly such confusion

could lead to an incorrect claim of the observation of leptonic CP-violation in a theory

which is CP conserving.

We recently performed a global analysis of oscillation data in the presence of NSI

relevant to neutrino propagation in matter in ref. [22]. For simplicity the analysis in

ref. [22] only constrained the CP conserving part of the Hamiltonian and for consistency

the observables most sensitive to CP violating eﬀects, i.e., ν

and ¯ν

appearance at LBL

experiments, were not included in the ﬁt. Consequently the issue of the possible confusion

between real NSI and leptonic CP-violation could not be addressed. Furthermore, under the

simplifying assumptions employed, the results of ref. [22] could not yield any conclusion on

the status of the determination of the ordering of the states in the presence of NSIs either.

In this paper we extend the analysis in ref. [22] to account for the eﬀect of NSI in

the observables sensitive to leptonic CP violation and to the mass ordering. Our goal is

to quantify the robustness of the present “hints” for these eﬀects in the presence of NSI

which are consistent with the bounds imposed by the CP-conserving observables. We start

by brieﬂy summarizing the formalism and notation in section 2. In doing so we take the

opportunity to present a parametrization-invariant formalism for leptonic CP violation in

the presence of a generalized matter potential induced by NSI. In section 3 we describe

the strategy employed in the study. Finally in section 4 we present the results, and in

– 2 –

JHEP06(2019)055

section 5 we summarize our conclusions. We present some detail of the construction of the

basis invariants for CP violation in appendix A.

2 Formalism

In this work we will consider NSI aﬀecting neutral-current processes relevant to neutrino

propagation in matter. The coeﬃcients accompanying the relevant operators are usually

parametrized in the form:

NSI

= −2

√

f,α,β

αβ

(¯ν

)(

fγ

f) ,

= −2

√

α,β

αβ

(¯ν

)

(

fγ

(2.1)

where G

is the Fermi constant, α, β are ﬂavor indices, and f is a SM charged fermion.

In this notation, ε

αβ

parametrizes the strength of the vector part of the new interactions

(which are the ones entering the matter potential) with respect to the Fermi constant,

αβ

∼ O(G

). In the second line we make explicit that, as in ref. [22], we assume

that the neutrino ﬂavor structure of the interactions is independent of the charged fermion

type, so that one can factorize ε

αβ

≡ ε

αβ

Ordinary matter is composed of electrons (e), up quarks (u) and down quarks (d). As

in ref. [22] we restrict ourselves to non-standard interactions with quarks, so that only ξ

and ξ

are relevant for neutrino propagation. A global rescaling of both ξ

and ξ

by a

common factor can be reabsorbed into a rescaling of ε

αβ

, therefore only the direction in

the (ξ

, ξ

) plane is phenomenologically non-trivial.

In this framework, the evolution of the neutrino and antineutrino ﬂavor state during

propagation is governed by the Hamiltonian:

= H

vac

+ H

mat

and H

¯ν

= (H

vac

− H

mat

)

∗

, (2.2)

where H

vac

is the vacuum part which in the ﬂavor basis (ν

, ν

) reads

vac

= U

vac

†

vac

with D

vac

diag(0, ∆m

, ∆m

) . (2.3)

Here U

vac

denotes the three-lepton mixing matrix in vacuum [1, 54, 55] which we

parametrize as [25], U

vac

= R

(θ

, 0) R

(θ

, 0) R

(θ

, δ

), where R

(θ, δ) is a com-

plex rotation in the ij plane

(θ, δ) =

cos θ sin θe

−iδ

−sin θe

iδ

cos θ

. (2.4)

This expression diﬀers from the usual one “U” (deﬁned, e.g., in eq. (1.1) of ref. [56]) by an overall

phase matrix: U

vac

= P U P

∗

with P = diag(e

iδ

, 1, 1). In the absence of non-standard interactions such

rephasing does not aﬀect the expression of the probabilities and produces therefore no visible eﬀect. In the

presence of NSI this holds under the conditions discussed in section 2.2.

– 3 –

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