energy-dependent neutrino mixing parameters at oscillation
TRANSCRIPT
FERMILAB-PUB-21-362-T, NUHEP-TH/21-04
Energy-Dependent Neutrino Mixing Parameters at Oscillation Experiments
K. S. Babu,1 Vedran Brdar,2, 3 Andre de Gouvea,2 and Pedro A. N. Machado3
1Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA2Northwestern University, Department of Physics & Astronomy,
2145 Sheridan Road, Evanston, IL 60208, USA3Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA
Neutrino mixing parameters are subject to quantum corrections and hence are scale dependent.This means that the mixing parameters associated to the production and detection of neutrinosneed not coincide since these processes are characterized by different energy scales. We show that,in the presence of relatively light new physics, the scale dependence of the mixing parameters canlead to observable consequences in long-baseline neutrino oscillation experiments, such as T2K andNOvA, and in neutrino telescopes like IceCube. We discuss some of the experimental signaturesof this scenario, including zero-baseline flavor transitions, new sources of CP-invariance violation,and apparent inconsistencies among measurements of mixing angles at different experiments oroscillation channels. Finally, we present simple, ultraviolet-complete models of neutrino masseswhich lead to observable running of the neutrino mixing matrix below the weak scale.
I. INTRODUCTION
The discovery of neutrino oscillations towards the end of the last century [1, 2] launched a diverse, world-wide experimental neutrino oscillation program that is expected to continue, at least, well into the nextdecade with the DUNE [3] and Hyper-Kamiokande [4] projects, currently under construction. It aimsat measuring, sometimes with exquisite precision, the neutrino oscillation phenomenon via a variety ofoscillation channels, baselines, and experimental conditions. The ultimate goal is to, broadly speaking,test the three-massive-active-neutrinos paradigm that postulates the existence of three neutral leptonswith different masses that interact only via the neutral-current and charged-current weak interactions,as prescribed by the Standard Model of particle physics (SM).
While the three-massive-active-neutrinos paradigm provides an excellent fit to virtually all neutrinodata,∗ the current data allow for the presence of more new physics in the neutrino sector. Different,well-motivated new physics scenarios can be probed by the current and next generation of neutrinoexperiments. These include the existence of new, light neutral-fermion degrees of freedom that mix withthe active neutrinos (“sterile neutrinos”), and new neutrino–matter interactions that manifest themselvesat the energies of interest via four-fermion operators (“non-standard interactions (NSI)”). The latter areusually associated to heavy new physics and are in general strongly constrained, in the absence of afair amount of fine-tuning, by charged-lepton processes [11–15], with some exceptions, see for exampleRefs. [16–19].
Here, we explore potential new phenomena associated to new, relatively light degrees of freedom thatinteract almost exclusively with neutrinos. In these scenarios, constraints from the charged-lepton sectorare significantly weaker while other constraints, including those associated to the existence of new, lightdegrees of freedom in the early universe, can be avoided. New interactions between neutrinos and new,light particles can impact neutrino experiments in two different ways: (i) the new states can be producedwhen neutrinos are produced or detected, leading to changes in the kinematics and flavor structureof neutrino scattering (see, for exampe, [20–25] and references therein) or (ii) quantum correctionsassociated to the virtual exchange of the new degrees of freedom can modify neutrino production anddetection. Here, we concentrate on the latter which, to the best of our knowledge, has not been exploredextensively in the literature.
At the core of the idea is the fact that, once higher-order corrections are included, the parameters thatdescribe neutrino oscillations are energy dependent. This is neither new nor surprising. For example,the renormalization-group (RG) running of neutrino oscillation parameters between the neutrino-mass-generating scale (often assumed to be much higher than the scale of electroweak symmetry breaking) andthe weak scale has been the subject of intense investigation in the literature, see for instance Refs. [26–33].
∗ There are experimental results that do not fit the three-massive-active-neutrinos paradigm including searches for electron(anti)neutrino appearance at the LSND [5] and MiniBooNE [6–8] experiments as well as the so-called reactor antineutrinoanomaly [9, 10]. Explanations to these remain elusive and will not be considered in any detail here.
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The running of mass and mixing parameters is not exclusive to the neutrino sector: quark masses runsignificantly above the GeV scale and the running of the bottom and the top masses has been directlyobserved at lepton and hadron colliders [34–39]. The CKM matrix elements are also expected to runabove the weak scale [40]; this running, however, has never been observed. Below the weak scale, thesematrix elements can be treated as constant in the absence of light new physics. This is usually a goodapproximation since new, light degrees of freedom that couple to quarks are strongly constrained.
Different from the quark sector, however, new light degrees of freedom that couple predominantlyto neutrinos are not strongly constrained. If these exist, RG running effects may be relevant in thecontext of neutrino oscillation experiments. In this paper, we show that neutrino oscillation probabilitiesare affected through the mismatch between the leptonic mixing matrix evaluated at the scale (or moreprecisely momentum transfer) corresponding to the neutrino production and the one at which neutrinosare detected. A careful treatment of the oscillation phenomenon, therefore, requires – for a fixed neutrinoenergy! – twice as many relevant mixing angles (production and detection values). The number of CP-oddparameters is also larger. We find that while running effects are already strongly constrained, they canimpact significantly the current and next generation of neutrino oscillation experiments, including T2Kand NOvA. Their presence may lead, for example, to apparent inconsistencies between measurements ofoscillation parameters between T2K and NOvA and between “accelerator” and “reactor” measurementsof oscillation parameters. On the other hand, there are simple new physics scenarios that lead tosignificant low-energy running of the oscillation parameters, including some that are related to the originof nonzero neutrino masses. Hence, these effects are not only possible in principle, they may be accessibleif neutrino masses are a consequence of relatively light, new physics.
This manuscript is organized as follows. In Section II, we set the stage by introducing and discussingthe general idea. Section III contains a detailed treatment of neutrino oscillation probabilities in thepresence of RG-running effects. First, in Section III A, we compute the vacuum oscillation probability ingeneral and discuss the more familiar, less cumbersome two-flavor case in some detail; in this simplifiedframework, we discuss the oscillation probabilities in different useful limits including circumstances whenthe running effects are small and the case in which the oscillation baseline is zero. We discuss the moreuseful but much more cumbersome three-flavor scenario concentrating on subsets of the parameter space.In Section III B, we discuss matter effects, which are relevant for the long baseline experiments underconsideration. For the two-flavor case, exact expressions are given while the realistic three-flavor casecan only be tackled, for all pragmatic purposes, numerically. In Section IV, we discuss two concretemodels that lead to large RG-running effects. A quantitative study of the consequences of these modelsis presented in Section V. There, we discuss some consequences for T2K and NOvA in light of constraintsfrom short-baseline experiments (Section V A). In Section V B, we scrutinize the impact on the flavorcomposition of ultra-high-energy neutrinos. Finally, we conclude in Section VI.
II. RUNNING OF NEUTRINO MIXING PARAMETERS
A simple way to understand that quantum corrections can lead to nontrivial effects in neutrino oscillationsis to investigate the charged-current weak interactions in the mass basis for both charged leptons andneutrinos. In more detail
− L ⊃ g√2Uαi ¯α /W
−PLνi +H.c. , (II.1)
where `α, α = e, µ, τ , are the charged-lepton fields, νi, i = 1, 2, 3, are the neutrino fields with welldefined masses m1,m2,m3, respectively, g is the SU(2)L gauge coupling, PL is the left-chiral projectionoperator, and Uαi are the elements of the leptonic mixing matrix. The product (gUαi) can be interpretedas the coupling between a W -boson, a charged lepton `α, and a neutrino νi. Once higher-order quantumeffects are included, the question of interest here is whether these allow (gUαi) to change relative to oneanother. When this happens, as we discuss carefully below, we can say that the mixing matrix “runs.”
It is easy to see that, ignoring fermion Yukawa coupling effects, higher order electroweak corrections to(gUαi) are trivially proportional to (gUαi): (gUαi)→ (gUαi)×F , where F does not depend on the indicesα or i. Fig. 1 (center-panel) depicts one of the many higher-order one-loop electroweak corrections, forillustrative purposes. The presence of new interactions changes the picture significantly as long as thesehave a nontrivial flavor structure. For example, a new interaction that involves only neutrinos and newdegrees of freedom, depicted schematically in Fig. 1 (right-panel), will modify the neutrino propagatorand, in turn, modify (gUαi)→ (gUαi)+
∑j(gUαj)×Fij . If the “loop-factors” Fij depend on i, j = 1, 2, 3,
the (gUαi) change in a flavor-dependent way.A simple, concrete model that would manifest itself in this way is adding a gauge-singlet scalar Φ to
3
W
∼ gUαi
νi `α
W
∼ g3Uαi
νi
W/Z
`α
W
∼ gUαjFij
νi
νj
`α
FIG. 1: Left: Leading order Feynman diagram for the W -boson, charged lepton `α, neutrino νi coupling. Center:Sample one-loop electroweak correction. This contribution is proportional to (gUαj)(gU
∗βj)(gUβi) = g3Uαi in the
limit where the fermion masses are negligible and U is unitary. Right: Sample one-loop correction from a newinteraction that modifies the neutrino propagator. This contribution is proportional to (gUαj)Fij .
the SM field content and allowing for neutrino–scalar Yukawa interaction of the form hijΦνiνj (afterelectroweak symmetry breaking). Starting at the one-loop level, it is easy to see how the dark circle inFig. 1 (right-panel) is realized. This is not the model we explore here. Instead, we will concentrate ontwo ultraviolet complete models,† introduced and discussed in more detail in Sec. IV.
Following the renormalization-group approach to capture the finite quantum corrections, when thedust settles, we assume that we can replace (gUαi) → (g(Q2)Uαi(Q
2)) in a way that both g and Udepend on the momentum-transfer (squared) Q2 that characterizes the interaction: both gauge couplingconstant and the elements of the mixing matrix “run.” Note that we are assuming that, for a fixedvalue of Q2, Uαi(Q
2) can always be expressed as elements of a unitary matrix so it is meaningful todiscuss U(Q2) as a running mixing matrix. When discussing neutrino production or detection, it is oftenconvenient to define the neutrino flavor eigenstates να. The discussion above implies that, given theexistence of the new, generation-dependent interactions, the neutrino flavor eigenstates are energy-scaledependent.
Neutrino flavor change as a function of the distance between source and detector depends on whichlinear combinations of νi couple to the different charged leptons for both the production and detectionprocesses – the flavor eigenstates mentioned above – and on the differences of the squares of the neutrinomasses. While quantum corrections also lead to running masses, in neutrino oscillations one is interestedin the pole masses.‡ The reason is we are interested in neutrinos that propagate a macroscopic distance.In the language of quantum field theory, the only contribution to the amplitude that characterizesneutrino production plus detection comes from on-shell neutrino exchange: the contributions from virtualneutrino exchange, to exquisite precision, cancel out.
In summary, RG effects impact neutrino oscillations in the sense that they render the neutrino mixingmatrix Q2-dependent. We discuss flavor-change in this context in great detail in the next section andreturn to concrete, phenomenologically-safe models that lead to nontrivial U(Q2) in Sec. IV.
III. NEUTRINO OSCILLATION PHENOMENOLOGY
As discussed above, the new interactions of interest imply that the neutrino charged-current-weak-interaction eigenstates νe, νµ, ντ depend on the scale of the neutrino-production and neutrino-detectionprocesses. We can choose the scale of the couplings to be the so-called Lorentz-invariant “momentumtransfer” Q2: if we adopt all momenta in a vertex to be incoming, then Q2 ≡ |(pν+p`)
2|, where pν and p`are, respectively, the momenta of the neutrino and charged lepton involved in the charged-current processof interest. For example, in π → µνµ we have Q2 = m2
π. In general, in a physics process characterized byQ2 where a charged lepton `α (α = e, µ, τ) is absorbed and a neutrino is produced, the coherent linearcombination of neutrino mass eigenstates νi (i = 1, 2, 3) produced is
να(Q2) = Uαi(Q2)νi. (III.1)
† The gauge-invariant realization of the Φνν operators is not renormalizable.‡ The running mass should be used in evaluating the neutrino production and detection processes. Neutrino masses,
however, are small enough that their impact is negligible in virtually all processes of interest.
4
We are interested in neutrinos that propagate a finite distance so, as discussed in the previous section,when referring to neutrino mass eigenstates, we always refer to on-shell mass eigenstates with on-shellmasses mi, that is νi ≡ νi(Q2 = m2
νi). Note that we assume neutrinos are always produced and detectedvia the charged-current weak interactions.
In vacuum, a neutrino mass eigenstate with energy E is an eigenstate of the propagation HamiltonianH. Its evolution, assuming that the neutrino is ultra-relativistic, has the familiar form
|νi(L)〉 = exp[−im2iL/2E]|νi(0)〉, (III.2)
where L is the distance propagated or the baseline. The Dirac bracket notation refers to the flavorwave-function of the neutrino state (e.g., a three-dimensional Hilbert space assuming there are threeindependent neutrino states). The probability Pαβ that a neutrino is produced associated to a chargedlepton `α in a process characterized by Q2
p and detected, a distance L away, in association with a charged
lepton `β in a process characterized by Q2d can be trivially, but carefully, computed:
Pαβ(L) =∣∣〈νβ(Q2
d)| exp[−iHL]|να(Q2p)〉∣∣2 =
∣∣∣∣∣∑i
Uβi(Q2d)U
∗αi(Q
2p) exp[−im2
iL/2E]
∣∣∣∣∣2
. (III.3)
If the kinematics of the production and detection processes are the same, we obtain the familiar vacuumneutrino oscillation expressions. In general, however, this is not the case. Imagine an experimental setupwhere neutrinos are produced in charged-pion decay: π+ → µ+ν. In this case, Q2
p = m2π, independent
from the pion and neutrino laboratory energies. Further imagine that the neutrino is detected viaν+n→ e−+p. In the neutron’s rest-frame, for large enough neutrino energies, Q2
d ∼ Emn; Q2d depends
on the neutrino detector-frame energy and, clearly, has no relation to the pion mass. In the subsectionsthat follow, we explore the consequences and subtleties of Eq. (III.3) and discuss how to handle thepropagation of neutrinos through matter when the mixing matrix is Q2 dependent.
Before proceeding, we wish to highlight that, for a fixed value of Q2p and Q2
d, the oscillation formalismwe will explore here is similar to what one would obtain when considering the hypothesis that there isnew physics in neutrino production and detection, and that the impact of new physics is process andflavor dependent [41] (for a more recent discussion, see also Ref. [42]). The two scenarios, however, arenot identical since, in the setup under consideration here, there are different mixing matrixes for differentneutrino-scattering energies, even if the physics processes in question are the same; our Uα depend onthe momentum transfer, not the nature of the neutrino-production and neutrino-detection processes. Asan aside, our discussion here is a little more complete relative to the one on Ref. [41] as we look intothree-flavor effects, matter effects, and new CP-violating phenomena more carefully.
A. Vacuum oscillations: two and more flavors
We first consider the simplified case of two charged-leptons and two neutrinos – e, µ, ν1, and ν2 forconcreteness – propagating in vacuum. The most general two-by-two Q2-dependent mixing matrix canbe expressed as
U(Q2) =
(1 0
0 eiγ(Q2)
)(cos θ(Q2) sin θ(Q2)− sin θ(Q2) cos θ(Q2)
)(eiα(Q2) 0
0 eiβ(Q2)
), (III.4)
where we indicate the Q2 dependency of the mixing parameters γ, θ, α, β explictly. Not all these pa-rameters are physical. We can redefine, with impunity, the kets |νe(Q2)〉 → exp[iζ(Q2)]|νe(Q2)〉 and|νµ(Q2)〉 → exp[iη(Q2)]|νµ(Q2)〉, for every value of Q2, such that two among three complex phases canbe removed. For example,
U(Q2) =
(cos θ(Q2) sin θ(Q2)− sin θ(Q2) cos θ(Q2)
)(1 0
0 eiβ(Q2)
), (III.5)
allows one to access the answer to all possible oscillation-related questions. Finally, we are allowed to alsophase-redefine the mass-eigenstate kets |νi〉 with impunity. Hence, we can choose β(Q2) in Eq. (III.5)to vanish at some fixed value of Q2. In the most general case, therefore, the oscillation probabilities inEq. (III.3) will depend on one mass-squared difference ∆m2 ≡ m2
2 −m21 and three mixing parameters:
θ(Q2p) ≡ θp, θ(Q2
d) ≡ θd, and β(Q2d)− β(Q2
p) ≡ β. (III.6)
5
Throughout, in order to unambiguously define the mass-eigenstates, we choose m2 > m1.A different subtlety lies in the physical range for the mixing parameters. Q2-dependent sign redefini-
tions of |νe〉 and |νµ〉 allow one to choose, for example, θ(Q2) ∈ [−π/2, π/2]. Sign redefinitions of themass eingenstates allow one to constrain θ to a specific quadrant (for example, θ(Q2) ∈ [0, π/2]) for somefixed value of Q2. Hence, if we choose θp to lie in the first quadrant, there is no guarantee that the samewill be true of θd.
Before proceeding, we wish to point out that the phase β(Q2) in Eq. (III.5) bears a strong resem-blance to the so-called Majorana phase. However, they are not the same and should not be confused.Majorana phases are only physical when neutrinos are Majorana fermions and only manifest themselvesin phenomena related, directly or indirectly, to lepton-number violation. The relative phase β manifestsitself in ordinary flavor-conversions and has nothing to do with lepton-number violation. It is physicalfor both Majorana and Dirac neutrinos. It is fair to ask about the origin and interpretation of thisCP-odd new physics parameter and why it impacts neutrino oscillations even when there are only twofamilies of leptons. One way to understand it is as follows. With two families and assuming the massiveneutrinos are Dirac fermions, the weak-interaction couplings and the fermion masses can be chosen real.In the model under consideration here, however, there are more flavor-dependent interactions. If thereare CP-violating couplings in the new physics sector, those will manifest themselves at higher-order incharged-current processes and can mediate CP-violating effects. In oscillation-language, these are pa-rameterized by the (running) CP-odd phase β introduced above. Incidentally, it is trivial to show usingEqs. (III.3) and (III.5) that the effect of β in the oscillation probabilities in vacuum is to “shift” theoscillation phase: ∆m2L/2E → ∆m2L/2E + β.
With all this in mind, for two-flavors, Eq. (III.3) can be expressed as
Peµ = Pµe = sin2(θp − θd) + sin 2θp sin 2θd sin2
(∆m2L
4E+β
2
), (III.7)
and
Pee = Pµµ = cos2(θp − θd)− sin 2θp sin 2θd sin2
(∆m2L
4E+β
2
). (III.8)
The equality of Pµe and Peµ and Pee and Pµµ is a consequence of the unitary evolution of the neutrinostates and the fact that |νe(Q2)〉 and |νµ(Q2)〉, for any Q2, are a complete basis for the two-dimensionalHilbert space.§ This is independent from the presence of the complex relative phase β. However,Pµe = Peµ does not mean that T -invariance is guaranteed. More carefully, Eq. (III.7) states thatP (νe(Q
2p) → νµ(Q2
d)) = P (νµ(Q2p) → νe(Q
2d)). T -invariance is the statement P (νe(Q
2p) → νµ(Q2
d)) =
P (νµ(Q2d) → νe(Q
2p)). P (νµ(Q2
d) → νe(Q2p)) is given by Eq. (III.7) with θd ↔ θp and β → −β so
T -invariance is violated if β 6= 0, π.For antineutrinos, Pαβ(θp, θd, β) = Pαβ(θp, θd,−β) so CP -invariance is also violated if β 6= 0, π. Note
that, in general, Pαα 6= Pαα, which could be interpreted as an apparent violation of CPT. However,this does not signal violation of the CPT-theorem since CPT-invariance implies P (να(Q2
p)→ να(Q2d)) =
P (να(Q2d)→ να(Q2
p)), which is satisfied.
For Q2d = Q2
p, we recover the well-known two-flavor oscillation-expressions. In general, however, thesituation is qualitatively different. For example, in the limit L→ 0, flavor is violated if either θp 6= θd orβ 6= 0. This is easy to understand. In either case, the mixing matrices are different at different values ofQ2 so the linear combinations of neutrinos that couple to e and µ are different: 〈να(Q2
d)|νβ(Q2p〉 6= δαβ .
These zero baseline effects, in practice, will constrain running effects to be relatively small, as we discussmore quantitatively in Sec. V. The complex relative phase β, in turn, leads to a phase shift in theoscillatory phenomenon. This leads, for example, to different behaviors of the oscillation probabilitiesat zero and small baselines. At zero baseline, a small β still induces an oscillation phase and thusPαγ = δαγ + O(β2). At small-but-finite baseline L, while the standard oscillation probability goes asP stdαγ ∼ δαγ+O(∆2
ij), a nonzero β would induce Pαγ ∼ O(β∆ij), where we have defined ∆ij ≡ ∆m2ijL/2E.
In the limit where the running effects are small, it is easier to appreciate analytically the impact of thenew physics effects. Assuming θd − θp = εθ +O(ε2θ) and β = εβ +O(ε2β), both εθ, εβ � 1 and unrelated
§ In the two neutrino framework, one can show that Pee + Peµ = 1 and Pee + Pµe = 1. These translate into Pµe = Peµ.
6
to one another,
Peµ = Pµe = ε2θ +O(ε4θ) +[sin2 2θd − sin 4θdεθ +O(ε2θ)
] [sin2
(∆m2L
4E
)+εβ2
sin
(∆m2L
2E
)+O(ε2β)
].
(III.9)and
Pee = Pµµ = 1−ε2θ+O(ε4θ)−(sin2 2θd − sin 4θdεθ +O(ε2θ)
) [sin2
(∆m2L
4E
)+εβ2
sin
(∆m2L
2E
)+O(ε2β)
].
(III.10)In the zero-baseline limit, the new-physics effects are O(ε2θ, ε
2β), quadratically small in the limit εθ, εβ � 1.
For a finite baseline, instead, the new-physics effects are O(εθ, εβ). For example,
Peµ = Pµe =(sin2 2θd − εθ sin 4θd
)sin2
(∆m2L
4E
)+εβ2
sin2 2θd sin
(∆m2L
2E
)+O(ε2θ, ε
2β , εθεβ). (III.11)
If the effects of mixing-angle running are not large, long-baseline experiments are, in some sense, moresensitive than short-baseline experiments.
The case of three charged-leptons and neutrinos – e, µ, τ , ν1, ν2, ν3 – is straightforward but morecumbersome. Taking advantage of the invariance of observables on the overall phases of |να(Q2)〉,α = e, µ, τ and |νi〉, i = 1, 2, 3, the most general 3×3 Q2-dependent mixing matrix can be parameterizedas
U(Q2) =
1 0 00 c23(Q2) s23(Q2)0 −s23(Q2) c23(Q2)
c13(Q2) 0 s13(Q2)e−iδ(Q2)
0 1 0
−s13(Q2)eiδ(Q2) 0 c13(Q2)
(III.12)
×
c12(Q2) s12(Q2) 0−s12(Q2) c12(Q2) 0
0 0 1
1 0 0
0 eiα(Q2) 0
0 0 eiβ(Q2)
, (III.13)
where cij(Q2), sij(Q
2) are short-hand for cos θij(Q2), sin θij(Q
2), respectively, for ij = 12, 13, 23. The
complex phases α(Q2) and β(Q2) can be chosen such that they vanish at some value of Q2.The vacuum oscillation probabilities will be given by Eq. (III.3) with the elements of the mixing matrix
as defined in Eq. (III.13). These will depend on the usual mass-squared differences ∆m231 ≡ m2
3 −m21
and ∆m221 = m2
2 − m21 (the third mass-squared difference is not independent, ∆m2
32 ≡ m23 − m2
2 =∆m2
31 −∆m221), six mixing angles θij(Q
2p), θij(Q
2d), ij = 12, 13, 23, two “Dirac” phases δ(Q2
p), δ(Q2d) and
two additional complex-phase-differences, α ≡ α(Q2d) − α(Q2
p) and β ≡ β(Q2d) − β(Q2
p). As in the twoneutrino case, α and β will induce a shift in the solar and atmospheric oscillation phases: ∆21 → ∆21 +α,∆31 → ∆31 +β, ∆32 → ∆32 +β−α. The mass eigenstates can be unambiguously defined in a variety ofways. Here, it pays off to adopt a definition that does not depend on the mixing matrix in order to avoida Q2-dependent definition. Concretely, we take the standard definition: m2
2 > m21 and |∆m2
31| > ∆m221.
∆m231 > 0 defines the ‘normal’ mass ordering (NO), ∆m2
31 < 0 the ‘inverted’ one (IO).¶
The expressions for the oscillation probabilities for three flavors in the Q2-dependent case are lengthy.We have nevertheless found several features in the zero-baseline limit and in the limit of small RG effectsthat turn out to be instructive. Let us first turn to the expressions in the zero baseline (L = 0) limit. Forsimplicity, we take θ13 = 0 and θ23 = π/4 at production and assume that the difference between variousparameters at different scales is small. We define εij ≡ θij(Q2
d)−θij(Q2p), εδ = δ(Q2
d)−δ(Q2p), εα = α, and
εβ = β. Note that |α|, |β| � 2π. Hence, εij is the amount the angle runs between momentum-transferscorresponding to production and detection. We omit the “production” subscript and thus it shouldbe understood that all angles in the expressions below correspond to the momentum scale of neutrinoproduction. Finally, below, we keep up to quadratic terms in all εs. This leads to the zero-baseline
¶ A different choice would have been |Ue1(Q2)|2 > |Ue2(Q2)|2 > |Ue3(Q2)|2. This clearly depends on Q2 and may lead toconfusion. There is nothing wrong, however, with using something like |Ue1(Q2 = 0)|2 > |Ue2(Q2 = 0)|2 > |Ue3(Q2 =0)|2.
7
survival probabilities
Pee '∣∣∣∣1− ε212 + ε213
2− ε2α
2s2
12 + iεαs212
∣∣∣∣2 ' 1− ε212 − ε213 −1
4ε2α sin2 2θ12, (III.14)
and
Pµµ =
∣∣∣∣1− 1
4
(ε212 + ε213 + 2ε223 + ε2αc
212 + ε2β − ε13εαsδ sin 2θ12 + 2ε12ε13cδ
)+i
2
(εαc
212 + εβ
) ∣∣∣∣2' 1− 1
4(εβ − εαc212)2 − ε223 −
1
2(εαc12s12 − ε13sδ)
2 − 1
2(ε12 + ε13cδ)
2, (III.15)
where cδ = cos δ and sδ = sin δ.On the other hand, zero-baseline appearance probabilities include
Pµe '1
2
∣∣∣∣ε13 + eiδ(ε12 + i εαc12s12
)∣∣∣∣2 ' 1
8ε2α sin2 2θ12 −
εαε13
2sin 2θ12sδ +
ε212 + ε213
2+ ε12ε13cδ,
Pµτ '∣∣∣∣ε23 +
i
2
(εαc
212 − εβ
) ∣∣∣∣2 = ε223 +1
4(εαc
212 − εβ)2. (III.16)
As already inferred in the case of two flavors, the RG effects at zero baseline appear at O(ε2) which isclear from Eqs. (III.14) to (III.16). Moreover, it is trivial to show that all asymmetries Pαβ − Pαβ areexactly zero at zero baseline (without any approximations).
For a finte baseline, even approximate expressions for the oscillation probabilities are rather lengthy(full expressions may be found in [43]) so here we focus on asymmetries. We keep terms linear in ε andup to order s2
13 or ∆m221/∆m
231, assuming that the oscillation phase is near the atmospheric maximum.
Moreover, for terms that are order ε we only keep terms that are at most linear in s13 or ∆m221/∆m
231.
The muon neutrino disappearance asymmetry in vacuum is, for example,
Pµµ − Pµµ '{−(εαc
212 − εβ) sin2 2θ23 + 8ε12c
213s13c23s
323sδ − εδs4
23 sin2 2θ13
}sin ∆31−
− ε13 sin 2θ12 sin 2θ23sδ(1 + s2
23 cos ∆31
)sin ∆21 . (III.17)
We will see later that θ12 typically runs more than other angles and phases in the scenarios we willstudy, so we call attention to the fact that the ε12 term in the asymmetry above is suppressed by s13.The dependence on εα and εβ is relatively large. As already stressed in the two flavor scenario, at finitebaseline there are effects already at O(ε); compare for instance Eq. (III.17) with Eqs. (III.14) to (III.16).
The electron neutrino disappearance asymmetry is given by
Pee − Pee ' (εβ − εδ) sin2 2θ13 sin ∆31 − εα(s2
12 sin2 2θ13 sin ∆31 − sin2 2θ12 sin ∆21
), (III.18)
This expression does not depend on ε12, ε13 and ε23 to order s213 or ∆21. The formulae for Pee and Pee,
however, do contain those: the effects cancel in the difference Pee − Pee. Hence, by studying differencesbetween electron neutrino and electron antineutrino disappearances, one can access RG induced effectson phases. As discussed, the apparent violation of CPT symmetry can be seen from Eqs. (III.17)and (III.18), since the differences between neutrino and antineutrino disappearance are in principle notzero. These differences are CP-odd, as they change sign under the reversal of the signs of all the phases.
The difference between electron neutrino and antineutrino appearance probabilities in a muon(anti)neutrino “beam” is somewhat lengthy. Concentrating on the dominant terms up to order s13
and dropping ∆21 terms multiplied by new physics contributions, we obtain
Pµe − Pµe ' −8J∆21 sin2
(∆31
2
)[1 +
(2
ε12
sin 2θ12+ εα
cδsδ
)cot(∆31/2)
∆21
], (III.19)
where J = c213s13c12s12c23s23 sin δ is the Jarlskog invariant [44, 45]. At long baseline experiments, wherethis channel matters the most, such expansion is reasonable and accurate (ignoring the matter effects).We see that in Eq. (III.19) there is also a term that is ε-independent; that is the standard CP violatingterm. Interestingly, although the terms containing ε12 and εα are enhanced by ∼ ∆31/∆21, at the peak ofthe “atmospheric” oscillation cot ∆31/2 ' cotπ/2 = 0, which suppresses the CP violating effect. This isparticularly pronounced for the T2K setup as will be seen in Section V. We also stress that in the δ → 0limit, in which there is no standard CP violation in the lepton sector, new RG induced CP violation is
8
still present and nonzero (εα term). In the future, comparing the amount of leptonic CP violation inDUNE and T2HK will allow one to probe this scenario thanks to the different neutrino energy spectraat the two experiments.
In order to highlight the CP-conserving contribution, the sum of the electron neutrino and electronantineutrino appearance probabilities, for ∆21 → 0, is
Pµe + Pµe = 2 sin2 2θ13s223
[1 + 2ε13
cos 2θ13
sin 2θ13− c23
s13s23(ε12cδ − εαc12s12sδ)
]sin2
(∆31
2
). (III.20)
The first term is the dominant component of the standard contribution. It is interesting to observe thatthe effect of new physics is enhanced by 1/s13 ∼ 7. A change in Pµe + Pµe can be compensated, in
this channel, by shifting the value of sin2 θ23 or sin2 2θ13. Therefore a mismatch between the θ23 valuemeasured in the νµ disappearance mode versus the νe appearance mode or a mismatch between θ13 valuesmeasured at reactor neutrino experiments and beam νe appearance are signatures of our scenario. FromEqs. (III.19) and (III.20), we can infer that the current measurements of electron neutrino appearanceby T2K [46] and NOvA [47] should already constrain ε12, ε13 and εα to be below, roughly, 10%.
We summarize the qualitative effects of the running of the mixing matrix on neutrino oscillationphenomenology and provide some of the most promising and direct experimental probes of this scenariobelow:
(i) In general, the mismatch between the production and detection mixing matrices affects all neutrinooscillation channels.
(ii) The effect of the phase differences, α and β (see Eq. (III.13)), is a shift of the solar (∆21) andatmospheric (∆31) oscillation phases, respectively.
(iii) Zero baseline transitions happen at second order in the new physics parameters. Nonetheless,searches for short-baseline oscillations provide good experimental probes of this scenario, particu-larly if performed at high neutrino energies, which makes the neutrino production and detectionscales more distinct.
(iv) The impact of the running of the mixing matrix on long-baseline oscillation probabilities is firstorder in the new physics parameters and thus the determination of the same oscillation parame-ters at different scales and the precise energy dependence of the oscillation probability curves arepromising venues for probing this scenario.
(v) Mixing-matrix running may lead to apparent CPT violation: Pαα−Pαα is, in general, nonzero fora finite baseline. CPT-symmetry is, of course, still conserved. These asymmetries, or perhaps theratios (Pαα−Pαα)/(Pαα+Pαα) could be powerful probes of mixing-matrix running, especially dueto possible cancelations of systematic uncertainties.
(vi) Mixing-matrix running may also affect appearance channels in CP-violating and CP-conservingways. Long-baseline experiments yielding different neutrino energy ranges, including NOvA, T2K,DUNE, and T2HK, could be sensitive to this scenario. Two experimental signatures stand outin the case of electron appearance: a mismatch between θ13 values measured at reactor and ac-celerator neutrino experiments, or a disagreement on the θ23 values measured in appearance anddisappearance modes in beam neutrino experiments. In fact, current NOvA and T2K data areexpected to be already sensitive to new sources of CP violation, potentially constraining ε12 andεα to be below 10% or so.
B. Matter effects: two and more flavors
Neutrino flavor-evolution is modified in the presence of matter. The Hamiltonian that describes flavor-evolution as a function of the baseline is
H =∑i
m2i
2E|νi〉〈νi|+A(L)|νe(Q2 = 0)〉〈νe(Q2 = 0)|, (III.21)
where A(L) =√
2GFNe(L) is the matter potential, GF is the Fermi constant, and Ne is the electronnumber-density of the medium as a function of the baseline. The matter potential is a coherent forwardscattering phenomenon where the neutrinos interact with the electrons in the medium at zero momentum
9
transfer. Hence, the νe interaction state of interest here is the one at Q2 = 0. H can be expressed in anybasis of the Hilbert space, as usual. Here, there are several tempting ones: the mass-eigenstate basis, theinteraction-basis “at production”, and the interaction-basis “at detection”. The mass-eigenstate basisis especially useful since it allows one to readily compute the flavor-evolution for arbitrary values of Q2
p
and Q2d. The fact that H depends on νe(Q
2 = 0) also induces a natural choice for the complex phases
α, β, defined in Eq. (III.13): α(Q2 = 0) = β(Q2 = 0) = 0.It is instructive to discuss the case of two flavors and a constant A(L) = A, which can be solved
analytically. Using Eq. (III.5) and defining θ0 = θ(Q2 = 0) and setting β(Q2 = 0) = 0, the eigenstatesof the propagation Hamiltonian are
|ν1M 〉 = cosω|ν1〉 − sinω|ν2〉, (III.22)
|ν2M 〉 = sinω|ν1〉+ cosω|ν2〉, (III.23)
where
sin 2ω =A
∆Msin 2θ0, (III.24)
cos 2ω =(∆−A cos 2θ0)
∆M, (III.25)
∆M =[(∆−A cos 2θ0)2 +A2 sin2 2θ0
]1/2, (III.26)
and ∆ = ∆m2/(2E). ∆M is the difference between the eigenvalues associated to |ν2M 〉 and |ν1M 〉; |ν2M 〉is associated to the larger eigenvalue when A > 0. We labelled the “matter mixing angle” ω in order toremind the reader that these states are expressed in the mass-basis, not, as one is most familiar, in theinteraction basis.
It is straightforward but rather lengthy to compute Peµ ≡ P (νe(Q2p)→ νµ(Q2
d)). Given a |νe(Q2p)〉 at
L = 0, the flavor-state-vector at L is
|ν(L)〉 = |ν1M 〉〈ν1M |νe(Q2p)〉+ |ν2M 〉〈ν2M |νe(Q2
p)〉e−i∆ML, (III.27)
and hence
Peµ =∣∣〈νµ(Q2
d)|ν1M 〉〈ν1M |νe(Q2p)〉+ 〈νµ(Q2
d)|ν2M 〉〈ν2M |νe(Q2p)〉e−i∆ML
∣∣2 . (III.28)
The Dirac brackets in Eq. (III.28) are
〈ν1M |νe(Q2p)〉 = cos θp cosω − sin θp sinωeiβp ,
〈ν2M |νe(Q2p)〉 = cos θp sinω + sin θp cosωeiβp ,
〈ν1M |νµ(Q2d)〉 = − sin θd cosω − cos θd sinωeiβd ,
〈ν2M |νµ(Q2d)〉 = − sin θd sinω + cos θd cosωeiβd . (III.29)
Note that since Pee + Peµ = 1, the survival probability can be obtained trivially from the appearanceone (Eq. (III.28) with help from Eqs. (III.29)).
For three flavors, one can approach the issue of matter effects following the same steps we outline abovefor two flavors. We especially highlight the usefulness of performing computations in the mass-eigenstatebasis. Analytic results, even if one is willing to make several different simplifying assumptions, are veryhard to come by and are not illuminating. In Sec. V, we compute oscillation probabilities includingrunning effects for different experimental setups. There, matter effects are always included and we donot make use of any approximate expressions; Eq. (III.21) is treated numerically.
IV. BENCHMARK MODELS
In Section II we discussed the concept of RG evolution in the low-energy neutrino sector without speci-fying a complete model. In order to discuss quantitative effects at neutrino experiments, however, it isuseful to discuss ultraviolet-complete frameworks. We will focus on models that address the neutrinomass puzzle and in which the RG effects come from a “secluded sector.” We concentrate on two simplemodels.
10
L N N L
ϕ
H1 H1
H2 H2
L N N L
H
ϕ
H
ϕϕ
N N N N
FIG. 2: Left : Feynman diagram associated to the generation of neutrino masses in a scotogenic-like neutrinomass model in which M ∝ YN (Model 1). Middle: Feynman diagram associated to the generation of neutrinomasses in an inverse seesaw model in which M ∝ Y −1
N (Model 2). Right: Representative Feynman diagramresponsible for the running of YN in both Model 1 and Model 2. See text for details.
Model 1. First, we consider a variation of the scotogenic model [48] with a U(1) lepton numbersymmetry. The two Higgs doublets of the scotogenic model, H1 and H2, have zero lepton number butare distinguished by a Z2 symmetry under which H1 is odd while H2 is even. As in the original scotogenic
model, lepton number and the Z2 symmetries allow the term (H†1H2)2 + h.c. in the scalar potential. Themodel also comprises three right-handed neutrinos NR, with lepton number +1, which are odd underZ2. Finally, we add a complex scalar singlet ϕ, which is even under Z2 and has lepton number −2. Thenew physics Lagrangian includes
− L(1)ν = LYνH1NR + ϕN c
RYNNR + h.c., (IV.1)
where Yν and YN are matrices in generation space. The scalar potential is such that ϕ and H2 developvacuum expectation values (vev), while H1 does not. The active neutrinos acquire Majorana masses, asdepicted in the left panel of Fig. 2. The vev of ϕ, vϕ ≡ 〈ϕ〉, in particular, breaks lepton number, but theZ2 symmetry remains unbroken. Therefore, the active neutrinos do not mix with the gauge-singlets NR.Breaking lepton number spontaneously would lead to a massless Goldstone boson, the Majoron [49]. Inprinciple, the mass of the Majoron could be made nonzero by soft lepton number breaking terms in thescalar potential, such as µ2|ϕ|2.
Without loss of generality, YN can be taken as real, positive, and diagonal. Yν is a generic complexmatrix. The neutrino mass matrix is given by
M ijν =
vϕ
8√
2π2
∑k
{(Y ikν Y jkν Y kN
) [ M2H
2M2H − (Y kNvϕ)2
ln2M2
H
(Y kNvϕ)2− M2
A
2M2A − (Y kNvϕ)2
ln2M2
A
(Y kNvϕ)2
]},
(IV.2)where MH,A are the masses of the inert neutral scalar and pseudoscalar, respectively. Here, we areinterested in light ϕ and NR, and thus vϕ � v ≡ 246 GeV. In the limit of small N masses, i.e.,
M iN = Y iNvϕ/
√2�MH,A, Eq. (IV.2) reduces to
Mν 'vϕ
16√
2π2YνYNY
Tν ln
M2H
M2A
. (IV.3)
The mixing matrix is defined via diagonalization Mdiagν = U†MνU
∗, where Mdiagν is diagonal and contains
the neutrino masses. We assume the inert doublet masses to be of order the weak scale so Yν does notrun at low energies and can be treated as a constant parameter. YN , however, is scale dependent forvalues of the energy scale that are above the mass of the ϕ,N .
Model 2. The second benchmark model we will focus on is a version of the type-I seesaw mechanism[50–55] where the right-handed neutrino Majorana masses arise from the (possibly explicit) breaking oflepton-number in the scalar potential. The model includes 3 SM singlet fermions NR with lepton number1, and a singlet scalar ϕ with lepton number −2. The Yukawa Lagrangian reads
− L(2)ν = LYνHNR + ϕN c
RYNNR + h.c., (IV.4)
where H is the SM Higgs boson. The vev of ϕ breaks lepton number.∗ The diagram generating neutrino
∗ As in Model 1, this would predict a Majoron. An active-neutrino–Majoron coupling would be induced. It, however, isdoubly suppressed by ν − N mixing and mν/vϕ, and could be very small (easily of order 10−9) for the right-handed
11
masses for this model is presented in the middle panel of Fig. 2. When ϕ develops a vev, active neutrinosacquire a mass matrix given by
Mν =
√2v2
4vϕYν(YN )−1Y Tν . (IV.5)
The mixing matrix is defined as before but now the functional dependence of U on YN is distinct fromthe previous model, changing the impact of YN running qualitatively with respect to Model 1.
For both models, it is straightforward to compute the scale dependence of YN . The right panel ofFig. 2 depicts a representative Feynman diagram that contributes to this running. The beta function ofYN can easily be calculated and yields, for both Model 1 and Model 2,
16π2β(YN ) ≡ 16π2 dYNd ln |Q|
= 4YN
[Y 2N +
1
2Tr(Y 2
N )
]. (IV.6)
Any relatively large entry in YN can lead to a significant running of all YN entries, and as a consequenceto observable running of the mixing matrix.YN is not directly related to the mixing matrix. In order to connect the running of YN and the running
of the mixing matrix, we need to specify other Lagrangian parameters, incluing the Yukawa couplingmatrices Yν (for both models). To achieve that, we make use of the Casas-Ibarra parametrization [56],which relates the Lagrangian parameters to the running leptonic mixing matrix and neutrino masses:
Yν =1√CY−x/2N R diag (mν1 ,mν2 ,mν3)U(Q2
p) , (IV.7)
where C are the vev-dependent prefactors in Eqs. (IV.3) and (IV.5) for Model 1, when x = 1, and Model2, when x = −1, respectively. R is, in general, a complex orthogonal matrix. In the next section, whencomputing new-physics effects in oscillation experiments, we have the freedom to choose values for theelements of R and the constants that make up C. We will assume R to be real and parameterized bythree Euler-like rotation angles ξ1 in 1-2 plane, ξ2 in 2-3 plane and ξ3 in 1-3 plane. The results we presentbelow are not qualitatively different if we are to assume ξ1, ξ2, ξ3 to be complex angles with magnitudesof order 1.
V. IMPACT ON NEUTRINO OSCILLATION EXPERIMENTS
Given the two concrete models outlined above, we proceed with the analysis of the impact of the runningof the mixing matrix elements at various neutrino experiments. Results for the two different models turnout to be qualitatively similar and the results presented below all correspond to Model 1. We take thisopportunity to stress one “advantage” of Model 1, namely the absence of mixing between active andsterile neutrinos. This makes it easier to satisfy experimental constraints on the mixing of the activeneutrinos with the relatively-light sterile neutrinos.
In Section V A, we discuss the signatures of the scenario at long-baseline experiments such as T2K[46, 57, 58] and NOvA [47, 59, 60] and then we confront these findings with the bounds from theshort-baseline experiments NOMAD [61–63], ICARUS [64], CHARM-II [65], and NuTeV [66, 67]. InSection V B, we focus on how this phenomenon impacts the flavor composition of astrophysical neutrinosmeasured by IceCube [68–70].
A. Long-Baseline Oscillation Experiments
We assume that the masses of the new-physics particles are of order the pion mass so RG running ofthe mixing parameters is only relevant for Q2 values larger than (100 MeV)2. For lower values of Q2,one can treat the mixing parameters as constant. The reason is we are mostly interested in neutrinosproduced at values of Q2
p ≤ m2π, since all beam neutrinos are predominantly produced in pion decay, and
hence, at Q2p, the mixing matrix is the same for all experimental setups. Furthermore, for reactor and
neutrino masses of interest. Hence the model is expected to be experimentally safe. Again, the Majoron could be givena nonzero mass via soft breaking terms in the scalar potential.
12
solar (anti)neutrino experiments, both Q2p and Q2
d are less than the pion mass-squared. Hence, we donot need to worry about running effects when it comes to extracting the current best-fit values of mostmixing parameters, as we discuss in more detail below.
Different values for Yν at Q2p are generated using Eq. (IV.7) and the following:
• We fix sin2 θ12(Q2p) = 0.310, sin2 θ13(Q2
p) = 0.022, ∆m221 = 7.53 · 10−5 eV2;
• We choose the atmospheric parameters θ23(Q2p) and ∆m2
31(Q2p) at random, with a flat prior on
their respective 3σ currently-allowed regions according to NuFIT [71]. The reason for this choiceis that atmospheric parameters are measured using experimental setups where Q2
d is larger thanm2π, hence we allow for a relatively large range of values at Q2
p;
• We choose the CP-odd phases α(Q2p), β(Q2
p) and δ(Q2p) at random, with a flat prior, from their full
allowed physical ranges;
• We fix the value of the lightest neutrino mass and the neutrino mass ordering. We will show resultsfor two values: 0.05 eV, marginally consistent with cosmological bounds on the sum of the activeneutrino masses [72], and 0.01 eV. We will also show results for both normal ordering (NO) andinverted ordering (IO). Note that, quantitatively, effects depend considerably on the mass orderingand the lightest neutrino mass. It is well known, for example, that the RG effects are strongest forquasidegenerate masses [32];
• We choose the angles that parameterize the orthogonal R matrix at random, with a flat prior, fromtheir full allowed physical ranges;
• We pick the Yukawa matrix YN , at Q2p, to be diag(0.2, 0.5, 0.7). We have checked that the results
of the scans are qualitatively independent from the choice of YN (Q2P ) as long as the couplings are
non-degenerate and large enough to induce significant RG running.
As an aside, we have checked that the aforementioned choices for the couplings do not lead to theappearance of Landau poles below 10 TeV.
For each mass matrix defined at Q2p as described above, we solve Eq. (IV.6) numerically and compute
the mixing matrix at the different relevant values of Q2d. With that information, we compute the oscil-
lation probabilities numerically, as discussed in detail in Sec. III, including matter effects. Throughout,we will use these randomly generated scenarios to discuss the reach of RG-running effects. A completescan of the parameter space is not practically feasible given its dimensionality.
At detection, neutrinos interact mostly with the nucleons in the detectors through t-channel vector-boson exchange; the associated Q2 can take any value in a continuous interval.† In order to estimatethe RG effects accurately, Q2
d values should be extracted in an event-by-event basis. Our goals inthis manuscript, however, are to illustrate the effects of the running of the mixing matrix in simplemodels that explain nonzero neutrino masses and to demonstrate that these can be observed in neutrinooscillation experiments. With this in mind, we take a simplified approach that should prove to be agood approximation statistically. In 2 → 2 scattering kinematics (see e.g. Ref. [73]) the minimal andmaximal values of t = −Q2 are fixed, and thus we associate the mean value of the two to −Q2
d. Thisyields Q2
d = (2mNE2)/(2E + mN ), where mN is the nucleon mass and E is the neutrino energy. We
take E to be the average neutrino energy in short-baseline experiments while for T2K and NOvA westudy the impact of the running for the peak of their respective energy spectra. Table I lists the relevantenergies and corresponding
√Q2d values for all experiments considered here.
The left panel of Fig. 3 depicts the bi-probability plot at T2K (red points) and NOvA (blue points)for the NO, assuming the lightest neutrino mass to be m1 = 0.05 eV, for 30, 000 values of the modelparameters, generated following the procedure described above. Note that here we do not take intoconsideration constraints from other experiments, to which we return momentarily. The panel alsodepicts the bi-probabilities accessible in the absence of RG running (green for T2K, yellow for NOvA),for values of the atmospheric parameters ∆m2
31 and sin2 θ23 picked at random (flat prior) from theirrespective 1σ currently-allowed regions, according to NuFIT [71] and for values of δ also chosen atrandom from δ ∈ [0, 2π]. The red and blue points with error bars represent the results of analyzing T2Kand NOvA data, respectively, adapted from Ref. [74]. Fig. 3 reveals that running effects can be very
† This is to be contrasted with what happens in s-channel scattering, where the Q2 is defined by the incoming neutrinoenergy and the mass or energy fraction of the target particle.
13
Experiment E (GeV)√Q2d (GeV) channel constraint
T2K [46, 57, 58] 0.6 0.56 – –
NOvA [47, 59, 60] 2.1 1.27 – –
ICARUS [64] 17 3.94 νµ → νe 3.4× 10−3
CHARM-II [65] 24 4.70 νµ → νe 2.8× 10−3
NOMAD [61–63] 47.5 6.64νµ → νe 7.4× 10−3
νµ → ντ 1.63× 10−4
NuTeV [66, 67] 250 15.30νµ → νe 5.5× 10−4
νe → ντ 0.1
νµ → ντ 9× 10−3
TABLE I: Reference values of neutrino energy and√Q2d for all experiments considered here. For T2K and NOvA
we take the peak of the neutrino spectra, while for all others (the short-baseline experiments) we use the averageneutrino energy. The last two columns summarize the short-baseline constraints imposed (not applicable for T2Kand NOvA). See text for details.
FIG. 3: Bi-probability plots for NO and m1 = 0.05 eV. Left : Red and blue points are for T2K and NOvA,respectively, for the case where RG effects are taken into account. These points are to be compared to therespective standard 1σ-allowed regions (green for T2K, yellow for NOvA). Right : Same as Left, except that onlypoints that satisfy the zero baseline constraints are included. See text for details.
significant; they lead to appearance probabilities that differ from “no-running” expectations by morethan an order of magnitude. Effects at NOvA are more pronounced because (a) the typical neutrinoenergies are larger at NOvA and the mixing parameters have more “room” to run and (b) at T2K,as discussed around Eq. (III.19), L/E values are such that CP-odd effects due to the new physics aresuppressed.
As discussed in Sec. III, RG running leads to zero baseline effects since, in general, U(Q2p)U
†(Q2d) 6= 1.
Therefore, experiments with very short-baselines, designed, with the benefit of hindsight, for probingneutrino scattering physics or light sterile-neutrino phenomenology, are sensitive to this type of newphysics. In particular, setups with high average neutrino energy are especially sensitive due to the largerdifference between Q2
p and Q2d and hence potentially stronger running effects.
We identified several short-baseline experiments that place stringent constraints on the running: NO-MAD [75, 76] (bounds from CHORUS [77, 78] are qualitatively comparable and so are the associatedneutrino energies), CHARM-II [79], ICARUS [80] and NuTeV [81, 82]; see Table I for their referencevalues of energies and momentum transfers. All these experiments measured neutrino beams which wereprimarily composed of muon neutrinos.‡ Electron-neutrino appearance was severely constrained by all
‡ As a simplifying assumption, we treat all beam neutrinos as if they were the product of pion decay in flight.
14
FIG. 4: Same as Fig. 3 for the case of NO and m1 = 0.01 eV.
of them, while NOMAD and NuTeV also constrained anomalous νµ → ντ transitions. NuTeV, due toits non-negligible beam-νe component, also managed to put a bound on νe → ντ appearance. Concreteupper bounds for each appearance channel are listed in Table I. While NOMAD and NuTeV provide themost stringent bounds on appearance probabilities, it is important to include constraints obtained withdifferent neutrino energies and thus the information provided by the other experiments is invaluable.
The right-hand panel of Fig. 3 depicts the bi-probability plot for T2K (red points) and NOvA (bluepoints) for the NO, assuming the lightest neutrino mass to be m1 = 0.05 eV. As before, we revisitthe same 30,000 model-points depicted in the left-hand panel of Fig. 3 but here only keep those pointsthat satisfy the constraints from the short-baseline experiments listed in Table I. Models associated toa very large effect on NOvA and T2K appearance channels were also very likely to violate the short-baseline constraints. From a model-parameter perspective, zero-baseline constraints disfavor regions ofthe parameter space with O(1) Yukawa couplings when the neutrino masses are large. Ultimately, thenew physics effects consistent with short-baseline constraints are, at T2K and NOvA “perturbations” onstandard oscillations. Notice, however, that significant deviations are still allowed; a significant fractionof the points in the right-hand panel of Fig. 3 lies outside the standard (no running) 1σ allowed rangesfor T2K and NOvA.
RG running effects depend strongly on the active neutrino masses and tend to be largest when theseare quasi-degenerate. The left panel of Fig. 4 depicts the bi-probability plot at T2K (red points) andNOvA (blue points) for the NO, this time assuming the lightest neutrino mass to be m1 = 0.01 eV, for30, 000 values of the model parameters. The right panel of Fig. 4 depicts the subset of points that satisfythe short-baseline constraints. Comparing Fig. 3 with Fig. 4, we see that the region of the bi-probabilityplots accessible to the new physics is larger for quasi-degenerate neutrinos, m1 = 0.05 eV, when comparedto the more hierarchical case, m1 = 0.01 eV.
Similarly, Fig. 5 depicts the RG effects on T2K and NOvA for the IO and m3 = 0.01 eV (the lightestneutrino mass). Here, even for a relatively light lightest neutrino mass, running effects are comparableto the NO scenario with larger neutrino masses, discussed earlier. Thus, for the same mass of the lightestneutrino, the RG effects are larger for IO relative to NO. As expected, the short baseline constraints arevery relevant here as well.
While Figs. 3 to 5 illustrate the general overall reach of RG effect on the appearance channels, it isalso useful to understand how RG effects impact oscillations for a specific fixed values of the oscillationparameters at Q2
p. Fig. 6 depicts the impact of RG running at T2K (left) and NOvA (right), includingconstraints from the short-baseline experiments, for different fixed values of all oscillation parametersat Q2
p. Modulo rare outliers, the relative new physics effect, that is, the distance from respective blackdot, is of order 10%. Moreover, the RG effects tend to modify neutrino and antineutrino appearance bysimilar amounts, while extra contributions to the CP asymmetry are somewhat suppressed. This canbe understood from Eq. (III.19), in particular the cot(∆31/2) term that is almost vanishing for T2K atthe peak of the energy spectrum. Similar shifts due to new physics on both neutrino and antineutrinoappearances can be mimicked by changing the value of sin2 θ23. Therefore, one possible experimentaloutcome of this scenario could be an apparent inconsistency between the sin2 θ23 values obtained fromthe disappearance channel and the sin2 θ23 from appearance channel.
Since RG effects modify oscillation probabilities at T2K and NOvA in slightly different ways, it isinteresting to investigate whether they could explain the mild tension between the current T2K and NOvA
15
FIG. 5: Same as Fig. 3 for the case of IO and m3 = 0.01 eV.
0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
P(νμ→νe)
P(ν
μ→ν e)
T2K
NO (solid), IO (dashed)
sin2θ23=0.55, Δm312 =2.56×10-3eV2, δ=-π/2
sin2θ23=0.55, Δm312 =2.56×10-3eV2, δ=0
sin2θ23=0.55, Δm312 =-2.46×10-3eV2, δ=π/2
0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
P(νμ→νe)
P(ν
μ→ν e)
NOvA
NO (solid), IO (dashed)
sin2θ23=0.58, Δm312 =2.52×10-3eV2, δ=-π/2
sin2θ23=0.58, Δm312 =2.52×10-3eV2, δ=0
sin2θ23=0.57, Δm312 =-2.41×10-3eV2, δ=π/2
FIG. 6: Bi-probability plots – T2K on the right, NOvA on the left – for both mass orderings and the lightest massset to 0.05 eV. The dots indicate the three benchmark points in the standard case. These lie on their respectiveellipses, obtained by fixing θ23 and ∆m2
31 and varying δ ∈ [0, 2π]. The colored points represent different choicesof the new-physics parameters and are indicative of typical RG effects once zero baseline constraints are included.
electron (anti)neutrino appearance data sets. To do this, we designed a toy χ2 using the aforementionedT2K and NOvA data points in the bi-probability plane,
χ2 =∑X
∑i=ν,ν
(DXi − TXiσXi
)2
, (V.1)
where X denotes T2K or NOvA, DXi (TXi ) is the measured (predicted) value of the probability at the
experiment X and run mode i = ν, ν, and σXi is the error bar read from the plot. Out of the 30,000generated model-points, we found O(10) parameter points for NO and m1 = 0.05 eV which satisfythe short-baseline constraints and provide a slightly better fit to data, though none are statisticallysignificant. Since the combined fit to T2K and NOvA data leads to a small preference for the invertedordering [74], it is harder to improve the fit in the IO case. Out of the 30,000 generated model-pointsfor the IO, we found no points which provide a better fit to data compared to standard oscillations.
To further illustrate the RG effects, especially their dependence on the neutrino energy, we depictthe electron (anti)neutrino appearance probabilities at T2K (top) and NOvA (bottom), in Fig. 7, forNO and m1 = 0.05 eV. Red and green curves correspond to the oscillation probabilities for neutrinosand antineutrinos, respectively, at T2K, while curves in shades of blue are constructed by adopting theNOvA far detector baseline. The vertical bands indicate the peak of the energy spectrum for each ofthe two experiments. The left-hand panels depict the oscillation probabilities for the benchmark point
16
0.02
0.04
0.06
0.08
0.10
0.12
0.14
P(ν
μ->ν e)
or
P(ν
μ->ν e)
NO m1=0.05 eV
BP1 (solid)standard (dashed)
P(νμ->νe) T2K
P(νμ->νe) T2K
T2
K
1 2 3 4 5
-0.4
-0.2
0.0
0.2
0.4
0.6
E [GeV]
(PB
SM-
PS
M)/
PS
M
0.02
0.04
0.06
0.08
0.10
0.12
0.14
P(ν
μ->ν e)
or
P(ν
μ->ν e)
NO m1=0.05 eV
BP2 (solid)standard (dashed)
P(νμ->νe) T2K
P(νμ->νe) T2K
T2
K
1 2 3 4 5
0
5
10
15
20
25
E [GeV]
(PB
SM-
PS
M)/
PS
M
0.02
0.04
0.06
0.08
0.10
0.12
0.14
P(ν
μ->ν e)
or
P(ν
μ->ν e)
NO m1=0.05 eV
BP1 (solid)standard (dashed)
P(νμ->νe) NOvA
P(νμ->νe) NOvA
NO
vA
1 2 3 4 5-0.2
-0.1
0.0
0.1
0.2
E [GeV]
(PB
SM-
PS
M)/
PS
M
0.02
0.04
0.06
0.08
0.10
0.12
0.14
P(ν
μ->ν e)
or
P(ν
μ->ν e)
NO m1=0.05 eV
BP2 (solid)standard (dashed)
P(νμ->νe) NOvA
P(νμ->νe) NOvA
NO
vA
1 2 3 4 5
0
2
4
6
8
10
E [GeV]
(PB
SM-
PS
M)/
PS
M
FIG. 7: Oscillation probabilities as a function of the neutrino energy at T2K (upper panels) and NOvA (lowerpanels) including (solid) RG-running effects or not (dashed). In the left-hand panels, the new-physics parametersare the ones that provide the best fit we found for the combined T2K and NOvA data. In the right-hand panels,the new-physics parameters are strongly excluded by short-baseline constraints. We assume the NO and thelightest neutrino mass is set to 0.05 eV. The vertical lines indicate the neutrino energies where the T2K andNOvA spectra are largest. The bottom portion of each panel depicts the relative differences between oscillationprobabilities with and without RG-running effects.
in our scan (denoted by BP1)§ which best fits T2K and NOvA data (solid), compared with the casewhere the mixing parameters are the same at Q2
p but there are no running effects (dashed). The lowerpart of each figure depicts the relative difference between these two hypotheses. While the differencesare small, the relative differences are of order 10% percent, growing as the neutrino energy grows dueto larger accessible values of the momentum transfer. The right-hand panels of Fig. 7 depict the caseof a different benchmark point, denoted BP2. BP2 is excluded by the short-baseline data. Nonetheless,it serves to illustrate the possible impact of RG effects on oscillation experiments and to highlight theimportance of the short-baseline experiments.
Fig. 8 depicts the RG evolution of the relevant mixing parameters for both benchmark points BP1 and
BP2. The left-hand panel depicts the values of the different parameters as a function of√Q2 while the
right-hand panel depicts the ratio between each parameter at Q2 relative to its value at Q2p. For BP1
§ For completeness and reproducibility, we provide the two benchmark points discussed here. They canbe obtained by using Eq. (IV.7) with the parameters (δ, α, β, ξ1, ξ2, ξ3, θ23, ∆m2
31/10−3eV2) equal to(3.71, 1.57, 2.37, 3.45, 1.51, 3.00, 0.88, 2.437) for BP1 and (1.18, 0.24, 1.64, 5.48, 2.076, 1.85, 0.86, 2.525) for BP2 (allangles and phases are in radians).
17
0.5 1.0 1.5 2.0 2.50
20
40
60
80
Q2 [GeV]
mix
ing
pa
ram
ete
r[°]
NO m1=0.05 eVBP1 (solid)BP2 (dashed )
θ12
θ13
θ23
δ/10
α /10
β/10
T2
K
NO
vA
0.5 1.0 1.5 2.0 2.50.4
0.6
0.8
1.0
1.2
Q2 [GeV]
X/X(Q
p2)
NO m1=0.05 eVBP1 (solid)BP2 (dashed )
θ12
θ13
θ23
δ
α
β
T2
K
NO
vA
FIG. 8: Left : RG evolution of the mixing parameters for the two new-physics parameter points in Fig. 7. Thesolid (dashed) lines are for the best-fit (strongly excluded) point. The vertical lines denote the peak-value of√Q2 at T2K and NOvA. Right : RG evolution of the mixing parameters, normalized by the respective values at
Q2p.
(solid), RG evolution yields few percent-level changes in the values of oscillation parameters. Runningeffects are more pronounced in the case of BP2 (dashed). The strongest effects are in running of θ12,which is very strongly impacted by RG effects. This large variation in θ12 disturbs all flavor transitions,and induces sizable zero-baseline effects as U(Q2
d)U†(Q2
p) strongly deviates from unity. Hence, thesepoints in model-space are strongly constrained by both short- and long-baseline experiments. In generalwe find that, when strong RG evolution effects are present, they typically first appear in θ12. It is wellknown [32] that the variation of θ12 relative to the other mixing angles θ13 and θ23 is enhanced by the ratiobetween atmospheric and solar mass-squared differences: |∆θ12/∆θ13|, |∆θ12/∆θ23| ∝ |∆m2
31/∆m221|.¶
Finally, since RG effects on short-baseline experiment can be sizable, one may be tempted to search forexplanations to the LSND [5] and/or MiniBooNE [7, 8] anomalies using this framework. In particular,we found plenty of points in model-space with zero-baseline appearance probability Pνµ→νe ' 10−3
at around Eν ' 0.4 GeV, hence qualitatively in agreement with the oscillation interpretation of theMiniBooNE data. However, once constraints from NOMAD and NuTeV are imposed, such points areruled out. This is mostly because RG effects are stronger at larger neutrino energies. We will explore theshort-baseline phenomenology of this framework and variations thereof in more detail in an upcomingmanuscript.
B. Ultra-High-Energy Neutrinos from the Cosmos
Neutrino oscillation experiments are not the only way to search for the RG running of neutrino mixingparameters. In particular, RG effects grow with Q2 and the IceCube experiment has detected neutrinosfrom extra-galactic astrophysical sources with laboratory energies up to the PeV-scale. This correspondto√Q2d ' 103 GeV, far above the corresponding values accessible to terrestrial experiments (see Table I).
Here we argue that precise measurements of the flavor-composition of ultra-high-energy (UHE) neutrinosat IceCube are also sensitive to the new physics effects discussed here.
In Section V A, we focused on RG effects for m2π ≤ Q2
d . (16)2 GeV2 and concluded that new-physicseffects were a perturbation over the standard expectation, particularly due to constraints from short-baseline experiments. Here, we will modify the RG running conditions and assume that the RG effects
only take place for√Q2 � 16 GeV; i.e., the new degrees of freedom have masses of order tens of
¶ In our phenomenological discussions of RG running effects at T2K and NOvA, we did not highlight the fact thatθ12-running effects are largest. The reason is that long-baseline experiments (and all other “high-energy” experiments,including measurements of the atmospheric neutrino flux) have limited ability to constrain the solar oscillation parameterssince, for the typical energies associated to these setups E/∆m2 is much larger than the relevant baselines. While higherprecision and more statistics are expected at DUNE and Hyper-Kamiokande, these will also have only limited ability toindependently measure θ12.
18
GeV. This assumption leads to no observable effects at solar-system neutrino experiments but allows forpotentially strong effects on measurements of the flavor composition of IceCube’s UHE neutrinos. For
the purpose of our calculations, we postulate that running starts at√Q2p = 16 GeV and the values of all
the parameters at√Q2 = 16 GeV are consistent with existing constraints from oscillation experiments.
It is important to note that UHE neutrinos are produced, for the most part, in the decays of pions,muons or perhaps neutrons so Q2
p is usually around or below m2π, independent from the neutrino energy.
Since the running only starts at the mass scale of new particles, calculating the running from m2π to
Q2d is equivalent to setting the production Q2
p to the mass-scale of new particles. As far as detection at
IceCube is concerned, we fix√Q2d = 103 GeV, for simplicity.
The IceCube collaboration has released results [83, 84] on the flavor composition of astrophysicalneutrinos, an observable that is also one of the pillars of the forthcoming IceCube-Gen2 upgrade [85, 86].The initial flavor composition (νe:νµ:ντ ) of O(PeV) neutrinos (note that, here, we do not distinguishbetween neutrinos and antineutrinos) from astrophysical neutrino sources is uncertain due to our poorknowledge of the nature of these sources and the mechanism behind neutrino production. We will considerseveral hypotheses. If the neutrinos are predominantly produced in the decays of high energy pions andin the decays of the pion-daughter muons, the flavor ratios at production would be (1:2:0) [87, 88]. If,on the other hand, muons from the pion decays lose most of the their energy before decaying (e.g., dueto interactions in a dense medium [89]), their decay products are not very energetic and the initial flavorcomposition would be (0:1:0) [90, 91]. If, on the other hand, neutron decays are the dominant sourceof UHE neutrinos, only electron antineutrinos would be produced at the source: (1:0:0) [92]. It turnsout that neutron decays as the dominant source of UHE neutrinos are disfavored at around the 68%CLby Icecube data [83, 84]. For completeness, one could also consider a pure tau neutrino initial flavorcomposition (0:0:1) [93–95] even if this is not expected to occur in any known astrophysical environment.
Once produced, the propagation of neutrinos to the Earth is subject to neutrino oscillations. Given thevery long baselines, it is safe to treat these neutrinos are incoherent superpositions of mass eigenstatesand the oscillation probabilities at Earth are baseline independent. In the standard three-neutrinoparadigm, the probability of producing an astrophysical neutrino of flavor α and detecting it with flavorβ, α, β = e, µ, τ , is given by
Pνα→νβ = Pνβ→να = δαβ − 2∑k>j
Re[U∗αkUβkUαjU
∗βj
]=
n∑j=1
∣∣Uαj∣∣2∣∣Uβj∣∣2 . (V.2)
In the presence of mixing-matrix running, these probabilities are
Pνα→νβ =
3∑j=1
∣∣Uαj(Q2p)∣∣2∣∣Uβj(Q2
d)∣∣2 , (V.3)
where we again stress that mixing matrix elements evaluated at the Q2p and Q2
d scales are potentiallydifferent. The expected flavor composition at the surface of the Earth is
Xβ =∑α
Pνα→νβXprodα , (V.4)
where Xprodα is the fraction of neutrinos of flavor α at production.
Fig. 9 depicts the flavor compositions in the ternary plots that are often employed for this type ofstudy [96]. Each panel corresponds to one of the production scenarios discussed above. Green regionsindicate the accessible range in the flavor triangle for standard neutrino oscillations, whereas blue scatterpoints represent different scenarios with RG effects included. The parameter-scan strategy is similar tothe one discussed in Section V A for both the standard scenario and the different new-physics models.In all panels, the new-physics effects increase the accessible region of the flavor-triangle when comparedto the standard case due to the mismatch between production and detection mixing matrix elements.The most striking effect can be seen in the upper-right-hand panel, associated to (1:0:0) (neutron decaysource) at production. As we discussed, while this type of production is disfavored by present data inthe absence of new physics, the situation is different once RG running effects are included; as can beseen in the figure, there are plenty of blue points safely located inside the 68% CL region.
For the preferred (1:2:0) and (0:1:0) scenarios, we also depict, in addition to present constraints, thefuture projections of IceCube-Gen2 [85]. The region defined by the blue scatter points exceeds thoseregions significantly. Hence, as the efforts in the field of neutrino astrophysics lead to the discovery ofpoint sources and illuminate the neutrino production mechanism, it will be possible to use the flavor-
19
FIG. 9: The relative flavor composition of UHE astrophysical neutrinos at the Earth for different choices ofthe relative flavor composition at the source. The green region corresponds to expectations from standard three-neutrino oscillations while the blue scatter points represent the potential effects of mixing-matrix running. Presentlimits from IceCube are shown as thick dashed (68% C.L.) and thick solid (95% C.L.) lines. For pion-decay anddamped-muon sources (left-top and left-bottom, respectively), we also show IceCube Gen-2 projections as thindashed lines (68% and 95% C.L.). See text for details.
FIG. 10: Same as Fig. 9, for all possible flavor-compositions (x ∈ [0, 1]), assuming the sources can only produceelectron and muon neutrinos.
20
composition observable to probe the presence of RG induced new physics effects with some precision.Finally, we also consider the more general production scenario in which the flavor composition is a genericmixture of electron and muon neutrinos. Fig. 10 depicts the flavor triangles for initial flavor ratios givenby (x:(1− x):0), x ∈ [0, 1], where we scan over all possible values of x ∈ [0, 1]. As in Fig. 9, it is clearlyvisible that mixing-matrix running may lead to strong effects on the flavor composition at the Earth. Wecan repeat this exercise for a most-general-source of the type (x(1− y):(1− x)(1− y):y), x, y ∈ [0, 1]. Inthis case, we also find that running effects can land well outside the region one can access in the contextof the standard scenario.
We conclude this subsection by summarizing the main effects induced by the running of the mixingmatrix:
• Even if the running of the mixing matrix starts at energy scales higher than those accessible tosolar-system neutrino experiments, the impact on the flavor composition of UHE neutrinos can bequite large.
• The main effect of the running is to enlarge the set of allowed values for the flavor-ratios atIceCube, for all production mechanism. A signature of this scenario is the measurement of aflavor composition of the UHE neutrino flux that is inconsistent with expectations from standardthree-neutrino oscillations.
VI. CONCLUSIONS
We considered, within the context of simple, ultraviolet-complete models of neutrino masses, the effectsof scale-dependent lepton mixing parameters at neutrino oscillation experiments. In this framework, themixing matrix at production and detection may be different, leading to rich and novel phenomenologicalconsequences. We identified several robust experimental signatures of this framework, including newsources of CP-invariance violation, apparent mismatches between θ13 measurements at reactor and beamexperiments, apparent mismatches between θ23 measurements in νµ disappearance and νe appearancechannels, zero-baseline flavor transitions νµ → ντ , νµ → νe and νe → ντ , and apparent CPT-invarianceviolation in the form P (να → να) 6= P (να → να) for α = e, µ, τ .
Taking current experimental constraints from short-baseline experiments into account, we showedthat the renormalization group evolution of the mixing parameters can induce observable effects at T2K,NOvA, and future long-baseline neutrino experiments. As a complementary probe to short- and long-baseline experiments, we also scrutinized effects at neutrino telescopes, in particular those related to theflavor composition of ultra-high-energy neutrinos.
Observable effects of RG running of the leptonic mixing matrix are a potential consequence of new,relatively light degrees of freedom and new neutrino interactions. As we demonstrated, the new inter-actions may be restricted, mostly, to the neutrino sector and hence are difficult to constrain outsideof experiments that involve flavor-resolved neutrino scattering. The effects discussed here are qualita-tively different from the more familiar nonstandard neutral-current-like neutrino interactions, which oftenmanifest themselves via modified matter effects or new interactions that impact neutrino production ordetection. The latter, in particular, may also be described using different leptonic mixing matrices atneutrino production relative to neutrino detection, but do not share the scale dependency of the RGrunning effects discussed here. New, light degrees of freedom can also be directly produced in neutrinoscattering. These effects are complementary to what we are discussing here, and have been explored,rather recently, in the literature.
ACKNOWLEDGEMENTS
We thank Bill Bardeen and Joachim Kopp for useful discussions. This work was supported in part bythe US Department of Energy (DOE) grant #de-sc0010143 and in part by the NSF grant PHY-1630782.Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. This project has received support from the European Union’s Horizon 2020 research andinnovation programme under the Marie Sk lodowska-Curie grant agreement No 860881-HIDDeN.
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