Comprehensive renormalization group analysis of the littlest

PHYSICAL REVIEW D 97, 075010 (2018)

Comprehensive renormalization group analysis of the littlest seesaw model
Tanja Geib1,* and Stephen F. King2,†
1Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany
2School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom
(Received 2 October 2017; published 6 April 2018)
We present a comprehensive renormalization group analysis of the littlest seesaw model involving two right-handed neutrinos and a very constrained Dirac neutrino Yukawa coupling matrix. We perform the first χ2 analysis of the low energy masses and mixing angles, in the presence of renormalization group corrections, for various right-handed neutrino masses and mass orderings, both with and without supersymmetry. We find that the atmospheric angle, which is predicted to be near maximal in the absence of renormalization group corrections, may receive significant corrections for some nonsupersymmetric cases, bringing it into close agreement with the current best fit value in the first octant. By contrast, in the presence of supersymmetry, the renormalization group corrections are relatively small, and the prediction of a near maximal atmospheric mixing angle is maintained, for the studied cases. Forthcoming results from T2K and NOνA will decisively test these models at a precision comparable to the renormalization group corrections we have calculated.
DOI: 10.1103/PhysRevD.97.075010

I. INTRODUCTION
Despite the impressive experimental progress in neutrino oscillation experiments, [1], the dynamical origin of neutrino mass generation and lepton flavor mixing remains unknown [2,3]. Furthermore, the octant of the atmospheric angle is not determined yet, and its precise value is uncertain. While T2K prefers a close to maximal atmospheric mixing angle [4], NOνA excludes maximal mixing at 2.6σ C.L. [5]. The forthcoming results from T2K and NOνA will hopefully clarify the situation. An accurate determination of the atmospheric angle is important in order to test predictive neutrino mass and mixing models. The leading candidate for a theoretical explanation of neutrino mass and mixing remains the seesaw mechanism [6–10]. However the seesaw mechanism involves a large number of free parameters.
One approach to reducing the seesaw parameters is to consider the minimal version involving only two righthanded neutrinos, first proposed by one of us [11,12]. In such a scheme the lightest neutrino is massless. A further simplification was considered by Frampton et al. [13], who
*[email protected] †[email protected]
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

assumed two texture 0’s in the Dirac neutrino mass matrix MD and demonstrated that both neutrino masses and the cosmological matter-antimatter asymmetry could be explained in this economical setup via the seesaw and leptogenesis mechanisms [14]. The phenomenology of the minimal seesaw model was subsequently fully explored in the literature [15–21]. In particular, the normal hierarchy case in the Frampton-Glashow-Yanagida model has been shown to be already excluded by the latest neutrino oscillation data [20,21].
An alternative to having two texture 0’s is to impose constraints on the Dirac mass matrix elements. For example, the littlest seesaw (LS) model consists of two righthanded (RH) neutrino singlets NaRtm and NsRol together with a tightly constrained Dirac neutrino Yukawa coupling matrix, leading to a highly predictive scheme [22–27]. Since the mass ordering of the RH neutrinos as well as the particular choice of the Dirac neutrino Yukawa coupling matrix can vary, it turns out that there are four distinct LS cases, namely cases A–D, as defined later. These four cases of the LS model are discussed in detail in the present paper. In particular, we are interested in the phenomenological viability of these four cases of the LS model defined at the scale of some grand unified theory (GUT) when the parameters are run down to low energy where experiments are performed.
A first study of the renormalization group (RG) corrections to the LS model was performed in [28]. The purpose of the present paper is to improve on that analysis and to focus on the cases where the RG corrections are the most

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TANJA GEIB and STEPHEN F. KING
important. We therefore briefly review the progress and limitations of the approach and results in [28]. In [28] the authors focused on analytically understanding the RG effects on the neutrino mixing angles for cases A and B in great detail and threshold effects were discussed due to two fixed RH neutrino masses, taken as 1012 and 1015 GeV, close to the scale of grand unified theories ΛGUT ¼ 2 × 1016 GeV [28].1
These analytical results were verified numerically. Furthermore, cases C and D were investigated numerically. However, the RG running of neutrino masses and lepton flavor mixing parameters were calculated at low energies, always assuming phenomenological best fit values at high energies, which was justified a posteriori by the fact that in most cases the RG corrections to the neutrino mass ratio2 as well as the mixing angles were observed to be rather small [28]. Such cases with small RG corrections lead to an atmospheric mixing angle close to its maximal value, which is in some tension with the latest global fits. To account for the running of the neutrino masses, Ref. [28] modified the Dirac neutrino Yukawa matrix by an overall factor of 1.25 with respect to the best fit values obtained from tree-level analyses. This factor was chosen based on scaling the neutrino masses for case A to obtain appropriate values at the electroweak (EW) scale, and subsequently used for all four LS cases. In other words, the numerical analysis of Ref. [28] chose input parameters that were extracted from a tree-level best fit, and adjusted them by an overall factor based on one specific case to include some correction for the significant running in the neutrino masses.
There are several problems with the above approach [28], as follows:
(i) The overall factor of 1.25 to the Dirac neutrino Yukawa matrix implies that only the running of the neutrino masses themselves is significantly affected by the choice of input parameters, while the neutrino mixing angles are still stable. Furthermore, it assumes that keeping the ratio of the input parameters unchanged when incorporating RG effects is reasonable. Both assumptions turn out to be incorrect.
(ii) Having modified the Dirac neutrino Yukawa matrix based on case A, Ref. [28] employs the same factor for cases B–D, although the running behavior can change fundamentally with the LS case.
1In this paper we run the parameters from ΛGUT ¼ 2×1016 GeV down to the low energy scale ΛEW ¼ 1000 GeV, both with and without supersymmetry (SUSY). This is justified in some SUSY GUTs of flavor, where the flavor breaking scale is close to the GUT scale (see e.g. [25]). It is convenient to take the high energy scale to be fixed in order to compare the effects of running with and without SUSY.
2This is not true for the neutrino masses m2 and m3. Their running is significant as demonstrated in Figs. 1–4 in Ref. [28].

PHYS. REV. D 97, 075010 (2018)

(iii) Most importantly, as mentioned above, the RG

running of neutrino masses and lepton flavor mixing

parameters were calculated at low energies, assum-

ing phenomenological best fit values at high ener-

gies. Clearly the correct approach would be to

perform a complete scan of model input parameters

in order to determine the optimum set of high energy

input values from a global fit of the low energy

parameters. This is what we do in this paper. As a consequence, the measure of the goodness-of-fit3

yields less than mediocre results for the input

parameters used in Ref. [28]: χ2A;BðΛEWÞ ≈ 50, and

χ

2 C;D

ðΛEW

Þ

≈

175.

In

comparison,

our

complete

scan

here reveals much improved best fit scenarios with

χ2AðΛEWÞ ¼ 7.1, χ2BðΛEWÞ ¼ 4.2, χ2CðΛEWÞ ¼ 3.2 and χ2DðΛEWÞ ¼ 1.5. In the present paper, then, we perform a detailed RG

analysis of the LS model, including those cases where the

RG corrections can become significant. As such it is no

longer sufficient to fix the input parameters by fitting to the

high energy masses and mixing angles. Consequently, we

perform a complete scan of model parameters for each case

individually, to determine the optimum set of high energy

input values from a global fit of the low energy parameters

which include the effects of RG running, and to re-assess

whether RG corrections might still be sufficient to obtain a

realistic atmospheric mixing angle. We find that the largest

corrections occur in the standard model (SM), although

we also perform a detailed analysis of the minimal supersymmetric standard model (MSSM)4 for various values of

tan β for completeness; however, since the RG corrections

there are relatively small, we relegate some of those results

to an appendix. In all cases we perform a χ2 analysis of the

low energy masses and mixing angles, including RG

corrections for various RH neutrino masses and mass

orderings.

The layout of the remainder of the paper is as follows.

In Sec. II we review the LS model and define the four

cases A–D which we analyze. In Sec. III we discuss

qualitatively the expected effects of RG corrections in

the LS models. We focus on some key features that will

help understand the findings in later sections, instead of

aiming at a complete discussion of the RG effects. In

Sec. IV we introduce the χ2 function that we use to analyze

our results. In Sec. V we discuss the SM results in some

detail, since this is where the RG corrections can be the

largest, serving to reduce the atmospheric angle from its

near maximal value at high energy to close to the best fit

value at low energy in some cases. Sec. VI discusses the

3Note that the goal is to minimize the value for χ2, as defined in e.g. Ref. [23].
4When we refer to the MSSM or SM we really mean the LS models with or without supersymmetry. We use this rather imprecise terminology throughout the paper.

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results for the RG analysis of the LS model in the MSSM. In Sec. VII we compare the MSSM results to those of the SM, and show that the RG corrections in the SM are more favorable. Section VIII concludes the paper. Appendix A introduces the notation needed to discuss benchmark scenarios for the LS model in the MSSM, and Appendix B displays tables with the results of all MSSM scenarios investigated.

II. LITTLEST SEESAW
The seesaw mechanism [6–10] extends the SM with a number of right-handed neutrino singlets NiR as
−Lm ¼ LLYlHER þ LLYνH˜ NR þ 1Nc MRNR þ H:c:; ð1Þ 2R
where LL and H˜ ≡ iσ2HÃ stand respectively for the lefthanded lepton and Higgs doublets, ER and NR are the righthanded charged-lepton and neutrino singlets, Yl and Yν are the charged-lepton and Dirac neutrino Yukawa coupling matrices, and MR is the Majorana mass matrix of righthanded neutrino singlets. Physical light effective Majorana neutrino masses are generated via the seesaw mechanism, resulting in the light left-handed Majorana neutrino mass matrix

mν ¼ −v2YνM−R1YTν :

ð2Þ

The LS extends the SM by two heavy right-handed
neutrino singlets with masses Matm and Msol and imposes constrained sequential dominance (CSD) on the Dirac
neutrino Yukawa couplings. The particular choice of structure of YAν ;B;C;D and heavy mass ordering MAR;B;C;D defines the type of LS, as discussed below. All four cases
predict a normal mass ordering for the light neutrinos with a massless neutrino m1 ¼ 0.
The basic starting point of LS models is to consider some
small family symmetry such as A4 which admits triplet representations. The family symmetry is broken by triplet flavons ϕi whose vacuum alignment controls the structure of the Yukawa couplings. To illustrate how this works, we
sketch a model, where the relevant operators responsible
for the Yukawa structure in the neutrino sector are

Λ1 H˜ ðL¯ · ϕatmÞNatm þ Λ1 H˜ ðL¯ · ϕsolÞNsol; ð3Þ

where L combines the SU(2) lepton doublets such that it transforms as a triplet under the family symmetry, while Natm; Nsol are the right-handed neutrinos NR and H is the electroweak scale up-type Higgs SU(2) doublet, the latter two being family symmetry singlets but distinguished by some additional quantum numbers. The right-handed neutrino Majorana superpotential is typically chosen to give a diagonal mass matrix,

MR ¼ diagðMatm; MsolÞ:

ð4Þ

The idea is that CSD(n) emerges from flavon vacuum
alignments in the effective operators involving flavon fields ϕatm, ϕsol which are triplets under the flavor symmetry and acquire vacuum expectation values that break the family symmetry. The subscripts are chosen by noting that ϕatm correlates with the atmospheric neutrino mass m3, ϕsol with the solar neutrino mass m2. CSD(n) corresponds to the choice of vacuum alignments,

001 hϕatmi ¼ [email protected] 1 CA;

011 hϕsoli ¼ [email protected] n CA

1 011 or hϕsoli ¼ [email protected] ðn − 2Þ CA;

ðn − 2Þ ð5Þ

n

where n is a positive integer, and the only phases allowed are in the overall proportionality constants. Such vacuum alignments are discussed for example in [26].
In the flavor basis, where the charged leptons and righthanded neutrinos are diagonal, the cases A and B are defined by the mass hierarchy Matm ≪ Msol, and hence Mˆ R ¼ DiagfMatm; Msolg, and the structure of the respective Yukawa coupling matrix,

0 0
Case A∶ YAν ¼ [email protected] a

beiη=2 1 nbeiη=2 CA

a ðn − 2Þbeiη=2

0 0

beiη=2 1

or Case B∶ YBν ¼ [email protected] a ðn − 2Þbeiη=2 CA

ð6Þ

a

nbeiη=2

with a, b, η being three real parameters and n an integer. These
scenarios were analyzed in [28] with heavy neutrino masses of Matm ¼ M1 ¼ 1012 GeV and Msol ¼ M2 ¼ 1015 GeV.
Considering an alternative mass ordering of the two heavy Majorana neutrinos, Matm ≫ Msol, and consequently Mˆ R ¼ DiagfMsol; Matmg, we have to exchange the two columns of Yν in Eq. (6), namely,

0 beiη=2

1 0

Case C∶ YCν ¼ [email protected] nbeiη=2 a CA

ðn − 2Þbeiη=2 a

0 beiη=2

1 0

or Case D∶ YDν ¼ [email protected] ðn − 2Þbeiη=2 a CA;

ð7Þ

nbeiη=2

a

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which we refer to as cases C and D. For Matm ¼ M2 ¼ 1015 GeV and Msol ¼ M1 ¼ 1012 GeV, both these cases were studied in [28].
We apply the seesaw formula in Eq. (2), for cases A–D
using the Yukawa coupling matrices YAν ;B in Eq. (6) with MAR;B ¼ diagðMatm; MsolÞ and YCν ;D in Eq. (7) with MCR;D ¼ diagðMsol; MatmÞ, to give (after rephasing) the light neutrino mass matrices in terms of the real parameters ma ¼ a2v2=Matm, mb ¼ b2v2=Msol with v ¼ 174 GeV,

00 0 01 mAν ;C ¼ [email protected] 0 1 1 CA

011 01
þ mbeiη[email protected] n

n ðn − 2Þ 1 n2 nðn − 2Þ CA; ð8Þ

ðn − 2Þ nðn − 2Þ ðn − 2Þ2

00 0 01 mBν ;D ¼ [email protected] 0 1 1 CA

011 01

ðn − 2Þ

1 n

þ mbeiη[email protected] ðn − 2Þ ðn − 2Þ2 nðn − 2Þ CA: ð9Þ

n nðn − 2Þ n2

Note the seesaw degeneracy of cases A and C and cases B and D, which yield the same effective neutrino mass matrices, respectively. Studies which ignore RG running effects do not distinguish between these degenerate cases. Of course in our RG study the degeneracy is resolved and we have to separately deal with the four physically distinct cases.
The neutrino masses and lepton flavor mixing parameters at the electroweak scale ΛEW ∼ Oð1000 GeVÞ can be derived by diagonalizing the effective neutrino mass matrix via

UνLmνUTνL ¼ diagðm1; m2; m3Þ:

ð10Þ

From a neutrino mass matrix as given in Eqs. (8) and (9), one immediately obtains normal ordering with m1 ¼ 0. Furthermore, these scenarios only provide one physical Majorana phase σ. As discussed above, we choose to start
in a flavor basis, where the right-handed neutrino mass
matrix MR and the charged-lepton mass matrix Ml are diagonal. Consequently, the Pontecorvo-Maki-NakagawaSakata (PMNS) matrix is given by UPMNS ¼ U†νL. We use the standard PDG parametrization for the mixing angles, and the CP-violating phase δ. Within our LS scenario, the standard PDG Majorana phase φ1 vanishes and −φ2=2 ¼ σ.
The low energy phenomenology in the LS model case A
has been studied in detail both numerically [22,23] and
analytically [24], where it has been found that the best fit to

PHYS. REV. D 97, 075010 (2018)
experimental data of neutrino oscillations is obtained for n ¼ 3 for a particular choice of phase η ≈ 2π=3, while for case B the preferred choice is for n ¼ 3 and η ≈ −2π=3 [22,26]. Due to the degeneracy of cases A and C and cases B and D at tree level, the preferred choice for n and η carries over, respectively. The prediction for the baryon number asymmetry in our Universe via leptogenesis within case A is also studied [25], while a successful realization of the flavor structure of Yν for case B in Eq. (6) through an S4 × Uð1Þ flavor symmetry is recently achieved in Ref. [26], where the symmetry fixes n ¼ 3 and η ¼ Æ2π=3.
With the parameters n ¼ 3 and η ¼ Æ2π=3 fixed, there are only two remaining real free Yukawa parameters in Eqs. (6) and (7), namely a, b, so the LS predictions then depend on only two real free input combinations ma ¼ a2v2=Matm and mb ¼ b2v2=Msol, in terms of which all neutrino masses and the PMNS matrix are determined. For instance, if ma and mb are chosen to fix m2 and m3, then the entire PMNS mixing matrix, including phases, is determined with no free parameters. Using benchmark parameters (ma ¼ 26.57 meV, mb ¼ 2.684 meV, n ¼ 3, η ¼ Æ2π=3), it turns out that the LS model predicts close to maximal atmospheric mixing at the high scale, θ23 ≈ 46° for case A, or θ23 ≈ 44° for case B [26], where both predictions are challenged by the latest NOνA results in the νμ disappearance channel [29] which indicates that θ23 ¼ 45° is excluded at the 2.5σ C.L., although T2K measurements in the same channel continue to prefer maximal mixing [30]. Since no RG running is included so far, cases C and D predict the same atmospheric angles upon inserting the benchmark parameters.
III. RGE RUNNING IN LITTLEST SEESAW SCENARIOS
Although the best-fit-input parameters in the present paper were determined by means of numerically solving the renormalization group equations (RGEs), we briefly recap some features of the LS RG running to facilitate comprehending the distinctive behavior of the different cases. This qualitative discussion is based on the more thorough analytical approaches in Refs. [28,31].
We switch from denoting the heavy right-handed neutrino masses by Matm; Msol to labeling them by M1, M2 to avoid mixing up the different cases and their opposite ordering of heavy neutrino masses. That is to say that irrespective of the case discussed, M2 always denotes the higher scale and M1 the lower.
For the LS, there are three different energy regimes of interest. Starting at the GUT scale, we can use the full theory’s parameters and RGEs to describe the evolution down to μ ¼ M2. At μ ¼ M2, the heavier NR is integrated out, and the light neutrino mass matrix as well as the RGEs have to be adapted. It is important to carefully match the full theory on the effective field theory (EFT) below the seesaw scale, denoted by EFT 1. Using the modified RGEs,

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the parameters are further evolved down to μ ¼ M1, where the remaining NR is integrated out, and the parameters of this intermediate EFT 1 are matched to the EFT
below M1, denoted by EFT 2. Once again, the light neutrino mass matrix along with the RGEs have to be determined anew. As we assume a strong mass hierarchy M2 ≫ M1, it is important to decouple the heavy neutrinos sub-
sequently, and describe the intermediate RG behavior
accordingly.
Taking a closer look at the highest regime, we specify the
LS input parameters at the GUT scale, and additionally

choose the flavor basis, i.e. both YlðΛGUTÞ and MRðΛGUTÞ are diagonal. For now, we are interested in the evolution of
the neutrino mixing parameters, which implies narrowly
watching how the mismatch between the basis, where the
charged-lepton Yukawa matrix Yl is diagonal, and the one where the light neutrino mass matrix mν is diagonal, unfolds. Consequently, we track the RG running of Yl and mν. Above the seesaw threshold μ ¼ M2, the evolution of the flavor structure of mν is mainly driven by YνY†ν. Consequently, the varying flavor structures of the Dirac neutrino Yukawa
matrix need to be examined more thoroughly.

(i) Case A: Whether we take the benchmark input parameters as stated in Sec. II or the global-fit parameters determined in Sec. IV, there is a hierarchy a ∼ Oð0.04Þ ≪ b ∼ Oð0.4Þ which allows for further simplification.

0 b2 YνY†ν ¼ [email protected] nb2

nb2

ðn − 2Þb2 1

0 b2 3b2 b2 1

a2 þ n2b2

a2 þ nðn − 2Þb2 CA⟶ n¼3;a≪[email protected] 3b2 9b2 3b2 CA:

ð11Þ

ðn − 2Þb2 a2 þ nðn − 2Þb2 a2 þ ðn − 2Þ2b2

b2 3b2 b2

Consequently, Ref. [28] only considers the dominant 9b2 term and thereby solves the simplified RGE for mν analytically. (ii) Case B: In analogy to case A, there is a hierarchy with respect to the input parameters a ∼ Oð0.04Þ ≪ b ∼ Oð0.4Þ.

0 b2

ðn − 2Þb2

nb2

1

0 b2 b2 3b2 1

YνY†ν ¼ [email protected] ðn − 2Þb2 a2 þ ðn − 2Þ2b2 a2 þ nðn − 2Þb2 CA⟶ n¼3;a≪[email protected] b2 b2 3b2 CA:

ð12Þ

nb2

a2 þ nðn − 2Þb2

a2 þ n2b2

3b2 3b2 9b2

Therefore, the simplified RGE of mν, which only takes the dominant (33)-entry into account, can be solved analytically.
(iii) Case C: Due to the opposite ordering of heavy neutrino masses, the hierarchy arising from either the benchmark or the global-fit input parameters is also reversed, namely a ∼ Oð1.2Þ ≫ b ∼ Oð0.01Þ.

0 b2 YνY†ν ¼ [email protected] nb2

nb2

ðn − 2Þb2 1

0 b2 3b2 b2 1

a2 þ n2b2

a2 þ nðn − 2Þb2 CA⟶ n¼3;b≪[email protected] 3b2 a2 a2 CA:

ð13Þ

ðn − 2Þb2 a2 þ nðn − 2Þb2 a2 þ ðn − 2Þ2b2

b2 a2 a2

Even when considering only the dominant contributions arising from a2, the resulting simplified RGE of mν cannot
be solved analytically anymore due to the nondiagonal elements strongly affecting the flavor structure of mν. (iv) Case D: In analogy to case C, there is a hierarchy to the input parameters a ∼ Oð1.2Þ ≫ b ∼ Oð0.01Þ.

0 b2

ðn − 2Þb2

nb2

1

0 b2 b2 3b2 1

YνY†ν ¼ [email protected] ðn − 2Þb2 a2 þ ðn − 2Þ2b2 a2 þ nðn − 2Þb2 CA⟶ n¼3;b≪[email protected] b2 a2 a2 CA:

ð14Þ

nb2

a2 þ nðn − 2Þb2

a2 þ n2b2

3b2 a2 a2

Thus, even the simplified RGE of mν turns out to be too involved to be solved analytically. Consequently, cases C and D are both investigated via an exact numerical approach in Ref. [28].

Note that, as apparent from the discussion below Eqs. (8) and (9), it is YAν YAν † ¼ YCν YCν † and YBν YBν † ¼ YDν YDν †. However, due to the inverted hierarchy with respect to a, b (stemming
from the inverted heavy neutrino mass ordering), different
entries dominate the RG evolution of mν, leading to different

RG running behavior. Thus, the degeneracy of the cases is resolved. This means that although (in case of starting from the same set of benchmark input parameters) the neutrino masses and mixing angles of cases A–D at the GUT scale are all identical, the running behavior of the mixing angles,

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which is mainly governed by YνY†ν, is quite different. Moreover, the discussion above uncovers a deeper connection among the cases A ↔ B and cases C ↔ D manifest
in the shared respective input parameter as well as the similar/same structure of YνY†ν dominating the running of mν.
Having determined mνðM2Þ from either the analytical or numerical RG evolution, we need to diagonalize the light
neutrino mass matrix. That way, we obtain not only the neutrino masses m2;3ðM2Þ but also the transformation matrix Uν. The latter in combination with the unitary transformation Ul, diagonalizing Yl, yields the PMNS matrix, and thereby the neutrino mixing parameters at the scale μ ¼ M2.
Thus, still within the high energy regime, we focus on
the charged-lepton Yukawa matrix. Since we are interested
in the flavor mixing caused by the running of Yl, flavorindependent terms are neglected. But besides that, the RGE
for Yl can be solved analytically without further simplifications, meaning that once again YνY†ν drives the flavor mixing. Finally, at μ ¼ M2, Yl is diagonalized by means of the unitary transformation Ul. Consequently, one would have all necessary parameters at hand to extract approx-
imations for the mixing angles; see Ref. [28].
Taking a closer look at the intermediate energy regime, M2 > μ > M1, we need to employ EFT 1 to describe the parameters and RG running. At the threshold μ ¼ M2, the effective light neutrino mass matrix can be written as

mðν2Þ ¼ v2ðκð2Þ þ Y˜ νM−1 1Y˜ Tν Þ;

ð15Þ

where κð2Þ ∝ Yˆ νM−2 1Yˆ Tν stems from decoupling the heavier right-handed neutrino with mass M2. The expression Y˜ ν (Yˆ ν) is obtained from Yν by removing the column corresponding to the decoupled heavy neutrino of mass M2 (the right-handed neutrino of mass M1). Please note that the two terms on the right-hand side of Eq. (15) are governed by
different RGEs, leading to so-called threshold effects. The RGEs of κð2Þ and Y˜ νM−1 1Y˜ Tν have different coefficients for the terms proportional to the Higgs self-coupling and gauge
coupling contributions within the framework of the SM
[31]. In combination with the strong mass hierarchy of the
heavy right-handed neutrinos, which enforces a subsequent
decoupling, the threshold effects become significant, and
thereby enhance the running effects on the neutrino mixing parameters.5 From the discussion in Ref. [28], we learn that
the threshold-effect-related corrections to the neutrino
mixing angles between M2 and M1 are dominated by an

5This can be understood by assuming that if the expression UT ðκð2Þ þ Y˜ νM−1 1Y˜ Tν ÞU is diagonal, then UT ðxκð2Þ þ x˜Y˜ νM−1 1Y˜ Tν ÞU is only diagonal for x ¼ x˜. Since this is not the case here, meaning the two terms scale differently, there is an additional “off diagonalness.”

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expression proportional to κð2Þ. Hence, we examine the combination Yˆ νM−2 1Yˆ Tν for the four cases.
(i) Case A:

0 b2 3b2 b2 1

Yˆ νM−2 1Yˆ Tν

¼

M−[email protected] 3b2
sol

9b2

3b2 CA:

ð16Þ

b2 3b2 b2

(ii) Case B:

0 b2

Yˆ νM−2 1Yˆ Tν

¼

M−[email protected]
sol

b2

b2 3b2 1 b2 3b2 CA:

ð17Þ

3b2 3b2 9b2

(iii) Cases C and D:

00 0 0 1

Yˆ νM−2 1Yˆ Tν

¼

M

−1 atm

[email protected]

0

a2

a2 CA:

ð18Þ

0 a2 a2

It is evident that the different order of the heavy neutrino decoupling once again evokes distinct flavor structures, thus demonstrating that the connection between cases A and B and cases C and D carries on to lower energy regimes as well. Note that, although the flavor structure of κð2Þ drives the mixing parameter’s running from threshold effects, its contribution comes with a suppression factor. Moreover, bare in mind that we only considered the threshold effects arising in EFT 1, but no further contributions from both neutrino and charged-lepton sector. These additional contributions may compete with the threshold effects in some cases, and lead to deviations from the similar features of cases A and B and cases C and D.
Going below the lower threshold, μ < M1, the running effects of the mixing angles become insignificant. This is not the case for the running of the light neutrino masses, which is too complicated to describe analytically in all regimes, and therefore was not discussed above. Nevertheless, there are a few details of the neutrino matrix running that we want to briefly mention: depending on the size of the Yν entries, the sign of the flavor-independent contribution to the RGE of mν can switch; and the coefficients of the flavor-dependent contributions for the SM and MSSM differ including a sign switch in some. As a consequence, a parameter can run the opposite direction for the framework of the SM in contrast to the MSSM. This feature is most apparent for the light neutrino masses that exhibit strong overall running in opposite directions when comparing the LS in the context of the SM and in the context of the MSSM. In order to access all parameters— neutrino masses, mixing angles and phases—at all scales,

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we turn to an exact numerical treatment using the
Mathematica package REAP [31]. There are two conclusions to be emphasized from the
discussion above. (i) Despite yielding identical neutrino masses and mixing parameters at the GUT scale [for identical input parameters (a, b)], cases A and C and cases B and D show fundamentally different running behavior. (ii) There is an intrinsic connection between the evolution of case A ↔ B (case C ↔ D) which is reflected in the parameter b (a) dominating the running as well as YνY†ν being mainly diagonal (being driven by the same block matrix). This distinction between case A and B versus case C and D properties becomes even more evident when taking a closer look at the energy regime M2 > μ > M1.

IV. THE χ 2 FUNCTION
In the following, we fix n ¼ 3 and η ¼ Æ2π=3. Consequently, there are only two free real parameters remaining to predict the entire neutrino sector. In order to find the best-fit-input parameters ma and mb while keeping η ¼ Æ2π=3 and n ¼ 3 fixed, we perform a global fit using the χ2 function as a measure for the goodness of fit [23],

χ2 ¼ XN PiðxÞ − μi2: ð19Þ

i¼1

σi

Here, we collect our model parameters in x ¼ ðma;mb;n;ηÞ, and predict the physical values PiðxÞ from the littlest seesaw model. The latter are compared to the μi that correspond to the “data”, which we take to be the global fit
values of [32],

μi ¼ fsin2θ12; sin2θ13; sin2θ23; Δm221; Δ231ð; δÞg: ð20Þ

Furthermore, σi are the 1σ deviations for each of the neutrino observables. In case the global fit distribution is Gaussian, the 1σ uncertainty matches the standard
deviation, which is the case for several of the neutrino
parameters depicted in Table I. However, there are a few
cases where the deviations are asymmetric. To obtain
conservative results, we assume the distribution surround-
ing the best fit to be Gaussian, and choose the smaller
uncertainty, respectively. That way, we slightly overestimate the χ2 values. Since the CP-violating phases δ and σ
are either only measured with large uncertainties or not at all, we define two different χ2 functions.
(i) χ2 for which N ¼ 5, i.e., δ is not included in
Eq. (20), (ii) χ2δ for which N ¼ 6, i.e., δ is included when
performing the global fit.

TABLE I. Best fit values with 1σ uncertainty range from global fit to experimental data for neutrino parameters in the case of normal ordering, taken from [32].

Parameter from [32]
sin2 θ12 sin2 θ13 sin2 θ23 Δm221 Δm231 δ

Best fit values Æ1σ
0.306þ−00..001122 0.02166þ−00..0000007755
0.441þ−00..002217 ð7.50þ−00..1179Þ10−5 eV2 ð2.524þ−00..004309Þ10−3 eV2
−99°þ−5591°°

A χ2 function is required to have a well-defined and generally stable global minimum in order to be an appropriate measure for the goodness of fit. This is the case for all CSD(n) models under the assumption that the sign of η is fixed [23]. From former analyses of the LS [23,28], we know in which ballpark the best fit values of ma;b are to be expected, respectively. That way, we can define a grid in the ðma; mbÞ-plane over which we scan— meaning that we hand over the respective input parameters x ¼ ðma; mb; n ¼ 3; η ¼ Æ2π=3Þ at each point of the grid to the Mathematica package REAP [31]. REAP numerically solves the RGEs and provides the neutrino parameters at the electroweak scale, i.e. the PiðxÞ in Eq. (19). The latter are used to determine how good the fit is with respect to the input parameters ðma; mbÞ by giving an explicit value for χ2ðδÞ. In the next step, we identify the region of the global χ2ðδÞ minimum, choose a finer grid for the corresponding region in the ðma; mbÞ-plane and repeat the procedure until we determine the optimum set of input values.
As we use the Mathematica package REAP [31] to solve the RG equations numerically, it is important to mention that the conventions used in REAP slightly differ from the ones discussed in Sec. I. First of all, with the help of Ref. [31], we can relate the two neutrino Yukawa matrices, which leads to Y˜ ν ¼ Y†ν. This needs to be taken into account when entering explicit LS scenarios into REAP. Secondly, note that REAP also uses the PDG standard parametrization which means that the mixing angles are identical to ours, and the Majorana phase is given by −φ2=2 ¼ σ. REAP uses δREAP ∈ ½0; 2π½ whereas we use δ ∈ ½−π; π½. Consequently, it is δ ¼ δREAP − 2π for δ REAP ∈ ½π; 2πŠ.
V. SM RESULTS
We investigate the running effects on the neutrino parameters m2, m3, ϑ12, ϑ13, ϑ23, δ and σ numerically by means of REAP [31]. Our analysis involves not only the four different cases A–D but also four settings for the heavy

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TABLE II. Case A Case B Case C Case D

Best fit values for SM cases A–D with varying right-handed neutrino masses.

Matm [GeV]
1010 1010 1012 1013
1010 1010 1012 1013
1012 1015 1015 1014
1012 1015 1015 1014

Msol [GeV]
1012 1015 1015 1014
1012 1015 1015 1014
1010 1010 1012 1013
1010 1010 1012 1013

ma [meV]
35.670 37.968 39.505 38.011
35.636 37.958 39.498 37.978
36.950 47.215 47.226 39.029
36.915 47.188 47.198 38.994

mb [meV]
3.6221 4.1578 4.1592 3.7985
3.6600 4.2020 4.2031 3.8377
3.4974 3.9735 4.1757 3.7492
3.5340 3.9885 4.1913 3.7843

χ2
11.778 7.16 772 7.14 042
10.7043
6.41 862 4.40 508 4.38 607 5.85 644
11.7597 3.24 554 3.23 646 9.88 824
6.40 423 1.4981 1.49 388 5.21 486

χ2δ
11.8275 7.18 596 7.15 869
10.7479
6.43 381 4.45 905 4.44 012 5.87 664
11.8094 3.31 094 3.30 174 9.93 932
6.41 938 1.52 676 1.52 265 5.23 251

RH neutrino masses, namely ðM2; M1Þ ¼ ð1012; 1010Þ; ð1015; 1010Þ; ð1015; 1012Þ; ð1014; 1013Þ. For each case and
RH mass setting, we furthermore perform vacuum stability
checks which validate all scenarios under consideration. As
we fixed two of the four input parameters of the LS, namely ðn; ηÞ, depending on the case, we minimalize χ2ðδÞ with respect to the free input parameters ðma; mbÞ. From the scan of the free input parameters, we determine the optimum set of ðma; mbÞ at the GUT scale, which are presented in Table II together with their corresponding χ2ðδÞ
values (obtained at the EW scale). Overall, it turns out that the values for χ2δ are only slightly inferior to the ones for χ2—by about a few percent at most—and both measures for
the goodness-of-fit point towards the same input values ðma; mbÞ. Thus, we refer to χ2 in the following discussion.
When comparing the different RH neutrino mass settings
for each case, respectively, there are several observations to
reflect about.
(i) The first and foremost observation is that the RH mass setting ð1015; 1012Þ makes for the best fit to the global fit values given in Table I for
each of the LS cases individually, closely followed by the mass setting ð1015; 1010Þ. The scenario ð1014; 1013Þ is already significantly poorer, and the goodness of fit further deteriorates for ð1012; 1010Þ. This shows that it is beneficial for the running effects to have M2 closer to the GUT scale. In addition, the mass of M1 barely—as long as it is still viable for a seesaw scenario—changes the outcome which is to say that the heavier of the
RH neutrinos plays the dominant role regarding
RG running behavior and the goodness of fit.
The detailed results for the RH mass setting

ð1015; 1012Þ are shown in Figs. 1 to 4. The results for the remaining three mass settings are displayed
in Tables V and VI.
(ii) For case A the best fit values for mb for mass settings ð1015; 1012Þ and ð1015; 1010Þ, which yield nearly identical χ2’s, are almost the same, while the ma differ notably. Furthermore, mb decreases with M2. The same is true for case B. For cases C and D,
respectively, it is the best fit values for ma that are almost identical for the comparatively good RH mass settings ð1015; 1012Þ and ð1015; 1010Þ, and mb that does vary. Moreover, ma lowers with M2. Recalling the qualitative discussion in Sec. III, these
observations can most likely be traced back to the deeper connection between case A ↔ B as wpeffilffilffifficffiffiaffiffiffisffieffiffi C ↔ D. For A ↔ B, the parameter b ∝ M2mb dominates the RG effects of the mixingpaffinffiffiffigffiffilffieffiffisffiffi,ffi whereas for C ↔ D, the parameter a ∝ M2ma does so. This already hints towards the overall
importance of the running of the mixing angles in
order to predict feasible neutrino parameters at the
EW scale, which we come back to when investigat-
ing the different LS cases. This line of reasoning also
explains the first observation, namely that the mass
of the heavier RH neutrino impacts the goodness-of-
fit predominantly.
(iii) Cases A and B yield a nearly identical input param-
eter ma for each RH neutrino mass setting individually, which hints towards yet another correlation
between cases A and B. The same holds true for
cases C and D with slightly more deviation in ma in comparison to case A ↔ B. For the input parameter mb, there does not seem to be a correlation between the different LS cases. While the discussion above did

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COMPREHENSIVE RENORMALIZATION GROUP ANALYSIS …

PHYS. REV. D 97, 075010 (2018)

FIG. 1. Case A—SM with Matm ¼ 1012 GeV and Msol ¼ 1015 GeV.

FIG. 2. Case B—SM with Matm ¼ 1012 GeV and Msol ¼ 1015 GeV.

feature equivalent RG behavior of two LS cases,
respectively, this observation shows a correlation with respect to the absolute value of ma. The reason behind this connection, however, proves more elusive

because ma is related to the lighter RH neutrino scale for cases A and B but to the heavier scale for cases C and D. Nevertheless, we return to discussing this
feature towards the end of this section.

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FIG. 3. Case C—SM with Matm ¼ 1015 GeV and Msol ¼ 1012 GeV.

FIG. 4. Case D—SM with Matm ¼ 1015 GeV and Msol ¼ 1012 GeV.

To emphasize the importance of performing global fits to
the experimental data at the EW scale for each LS case separately, we compare the χ2 values of the modified benchmark scenarios from Ref. [28] with the best fit

scenarios obtained from our analysis. As already mentioned in the Sec. I, the input values ðma; mbÞ in Ref. [28] are taken from a tree-level best fit, and adjusted by an overall
factor of 1.25, which was obtained from case A and aims at

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