Abstract
Neutrinos are the most elusive particles in our Universe. They have masses at least one million times smaller than the electron mass, carry no electric charge and very weakly interact with other particles, meaning that they are rarely captured in terrestrial detectors. Tremendous efforts in the past two decades have revealed that neutrinos can transform from one type to another as a consequence of neutrino oscillations—a quantum mechanical effect over macroscopic distances—yet the origin of neutrino masses remains puzzling. The physical evolution of neutrino parameters with respect to energy scale may help elucidate the mechanism for their mass generation.
Introduction
Ever since their discovery in the 1950s (ref. 1), neutrinos have continued to surprise us. In the Standard Model (SM) of elementary particle physics, neutrinos are massless particles. However, since the results from the SuperKamiokande experiment in 1998 (ref. 2), the phenomenon of neutrino oscillations has been well established, indicating that neutrinos do have nonzero and nondegenerate masses and that they can convert from one flavour to another^{3}. This important result was followed by a boom of results from several international collaborations. Certainly, these results have pinned down the values of the various neutrino parameters to an incredible precision, especially considering that neutrinos are extremely elusive particles and the corresponding experiments are extraordinarily complex^{4}. Currently operating experiments and future investigations under construction are aimed at determining the missing neutrino parameters, such as the CPviolating phase (which can be important for understanding the matter–antimatter asymmetry in the Universe), the sign of the large masssquared difference for neutrinos, and the absolute neutrino mass scale. In addition, the cubic kilometre scale neutrino telescope at the South Pole, IceCube^{5}, has been successfully constructed to search for ultrahigh energy astrophysical neutrinos, while a number of underground experiments are looking for neutrinoless doublebeta decay (see refs 6, 7, 8, 9, 10) and others are waiting for neutrino bursts from galactic supernova explosions (see refs 11, 12).
However, the origin of neutrino masses and lepton flavour mixing remains a mystery, and calls for new physics beyond the SM. It is believed that new physics should appear somewhere above the electroweak scale (that is, Λ_{EW}~10^{2} GeV) but below the Planck scale (that is, Λ_{P}~10^{19} GeV) for the following reasons. First, the smallness of neutrino masses can be ascribed to the existence of superheavy particles, whose masses are close to the grand unified theory (GUT) scale (for example, Λ_{GUT}~10^{16} GeV), such as righthanded neutrinos in the canonical seesaw models^{13,14,15,16,17}. Moreover, the outofequilibrium and CPviolating decays of heavy righthanded neutrinos in the early Universe can produce a lepton number asymmetry, which will be further converted into a baryon number asymmetry^{18}. Therefore, the canonical seesaw mechanism combined with socalled leptogenesis provides an elegant solution to the generation of tiny neutrino masses and the matter–antimatter asymmetry in our Universe. Second, the strong hierarchy in chargedfermion masses (that is, , and ) and the significant difference between quark and lepton mixing patterns (that is, three small quark mixing angles while two large and one small leptonic mixing angles) could find their solutions in the framework of GUTs extended by a flavour symmetry^{19,20}. Therefore, an attractive and successful flavour model usually works at a superhighenergy scale, where quarks and leptons are unified into the same multiplets of the gauge group but assigned into different representations of the flavour symmetry group. Third, the SM Higgs particle with a mass of 126 GeV has recently been discovered at the Large Hadron Collider at CERN in Geneva, Switzerland^{21,22}. If this is further confirmed by future precision measurements and the topquark mass happens to be large, the SM vacuum will become unstable around the energy scale 10^{12} GeV. In this case, new physics has to show up to stabilize the SM vacuum^{23}. In the canonical seesaw model with heavy righthanded neutrinos, the SM vacuum is actually further destabilized. However, if an extra scalar singlet is introduced to generate righthanded neutrino masses, the SM vacuum can be stabilized and the tiny neutrino masses are explained via the seesaw mechanism^{24}. Hence, the assumption that neutrino masses and lepton flavour mixing are governed by new physics at a superhighenergy scale is well motivated.
The experimental results will guide us to the true theory of neutrino masses, lepton flavour mixing and CP violation. At the same time, they will also rule out quite a large number of currently viable flavour models. However, this will only be possible if the renormalization group running of neutrino parameters, which describes their physical evolution with respect to energy scale, is properly taken into account. Thus, it may help to elucidate the mechanism for neutrino mass generation. The aim of this Review is to examine neutrino renormalization group running in more detail. First, we briefly summarize the current status of neutrino parameters and the primary goals of future neutrino experiments, and present a general discussion about the effective theory approach and renormalization group equations (RGEs) in particle physics. Then, we consider several typical neutrino mass models in the framework of supersymmetric and extradimensional theories, and the running behaviour of neutrino parameters is described and explained. Finally, the impact of renormalization group running on flavour model building and leptogenesis is illustrated and emphasized.
Neutrino parameters at low energies
Neutrinos are produced in beta decay of radioactive nuclei, nuclear fusion in the Sun, collisions between nucleons in the Earth atmosphere and cosmicray particles and in the manmade highenergy accelerators. Since they are always accompanied by the charged leptons e, μ and τ in production, it is convenient to define the neutrino flavour eigenstates {ν_{e}›,ν_{μ}›,ν_{τ}›} and discriminate them according to the corresponding charged leptons. The neutrino flavour eigenstates ν_{α}› (for α=e,μ,τ) are related to three neutrino mass eigenstates {ν_{1}›,ν_{2}›,ν_{3}›} with definite masses {m_{1},m_{2},m_{3}} by the superposition ν_{α}›=U_{α1}ν_{1}›+U_{α2}ν_{2}›+U_{α3}ν_{3}›, where the 3 × 3 unitary matrix U is the socalled lepton flavour mixing matrix^{25,26,27}. It is conventional to parameterize U by three Eulerlike mixing angles {θ_{12},θ_{13},θ_{23}} and three CPviolating phases {δ,ρ,σ}, namely^{3}
with c_{ij}≡cos θ_{ij} and s_{ij}≡sin θ_{ij} for ij=12, 13, 23. As a consequence of quantum interference among the three neutrino mass eigenstates, neutrinos can transform from one flavour to another, when propagating from the sources to the detectors. This phenomenon of neutrino flavour oscillations will be absent if either the two independent neutrino masssquared differences and (or ) or the three leptonic mixing angles {θ_{12}, θ_{13}, θ_{23}} are vanishing. Note that we will use instead of .
Thanks to a number of elegant experiments in the past two decades^{3}, the phenomenon of neutrino flavour oscillations has now been firmly established. The latest global analysis of data from all existing past and present neutrino oscillation experiments provides our best knowledge on the neutrino mixing parameters, as shown in Table 1. Note that has been used in ref. 28 to fit the oscillation data in both cases of normal neutrino mass hierarchy (that is, m_{1}<m_{2}<m_{3}) and inverted neutrino mass hierarchy (that is, m_{3}<m_{1}<m_{2}), only the results from ref. 28 are listed in this table to get a ballpark feeling of the values of the neutrino parameters. Two other independent globalfit analyses in refs 29, 30 yield different bestfit values. However, the 3σ confidence intervals of neutrino parameters from all three groups are indeed consistent.
At present, although there are weak hints for a nonzero Dirac CPviolating phase δ (see the last row of Table 1), it is fair to say that no direct and significant experimental constraints exist for the leptonic CPviolating phases. Furthermore, since neutrino oscillation experiments are blind to the Dirac or Majorana nature of neutrinos and to the Majorana CPviolating phases {ρ,σ}, it is still an open question whether neutrinos are Dirac or Majorana particles. In the latter case, neutrinos are their own antiparticles, which would lead to neutrinoless doublebeta decay of some nuclear isotopes and can hopefully be confirmed with this kind of experiments^{31}. The primary goals of ongoing and forthcoming neutrino oscillation experiments are to precisely measure the three leptonic mixing angles, to determine the neutrino mass hierarchy, and to discover the leptonic Dirac CPviolating phase. In addition, nonoscillation neutrino experiments aim to pin down the absolute neutrino masses and to probe the Majorana nature of neutrinos.
Confronting theories with experiments
Although most neutrino parameters have already been measured with a reasonably good precision, the origin of tiny neutrino masses and bilarge lepton flavour mixing remains elusive. To accommodate tiny neutrino masses, one may have to go beyond the SM at the electroweak scale and explore new physics at a superhighenergy scale. In this case, an immediate question is how to compare theoretical predictions at a highenergy scale with the observables at a lowenergy scale. With this question in mind, we present a brief account of effective theories and renormalization group running, and describe how neutrinos fit into this framework.
Effective theory approach
The effective theory approach is very useful, and sometimes indispensable in particle physics, where interesting phenomena appear at various energy scales. The basic premise for this approach to work well is that the dynamics at lowenergy scales (or large distances) does not depend on the details of the dynamics at highenergy scales (or short distances). For instance, the energy levels of a hydrogen atom are essentially determined by the finestructure constant of the electromagnetic interaction α≈1/137 and the electron mass m_{e}≈0.511 MeV. At this point, we do not need to know the inner structure of the proton, and the existence of the top quark and the weak gauge bosons. That is to say, the energy levels of a hydrogen atom can be calculated by neglecting all dynamics above the energy or momentum scale Λ much higher than αm_{e}, and the corresponding error in the calculation can be estimated as αm_{e}/Λ. If a higher accuracy is required, Λ will increase and the dynamics at a higherenergy scale may be needed. See refs 32, 33 for general reviews on effective field theories.
Now, consider a toy model with a light particle φ and a heavy one Φ, whose masses are denoted by m and M, respectively. Since , there exist two widely separated energy scales. The Lagrangian for the full theory can be written as , where the interaction between the light and heavy particles has been included in the second term. Since we are interested in physical phenomena at a lowenergy scale , where the experiments are carried out, we can integrate out the heavy particle as in the pathintegral formalism. Hence, an effective Lagrangian involving only the light particle is derived, and higherdimensional operators appear in , where c_{i} is a coefficient and d_{i} stands for the mass dimension of . It is evident that the dynamics at the highenergy scale can affect the lowenergy physics by modifying the coupling constants and imposing symmetry constraints, but the overall effects are suppressed by the heavy particle mass M. The method of effective field theories becomes indispensable when we even do not know at all whether a complete theory with the heavy particles exists or not.
Matching and threshold effects
Ultraviolet divergences appear in quantum field theories if radiative corrections are taken into account. In the presence of higherdimensional operators, effective theories are nonrenormalizable in the sense that an ultraviolet divergence cannot be removed by a finite number of counter terms in the original Lagrangian. However, since there is an infinite number of higherdimensional operators in the Lagrangian _{eff}, it is always possible to absorb all divergences and obtain finite results with a desired accuracy. Although any physical observables should be independent of the renormalization scheme used, it is a nontrivial task to choose a convenient renormalization scheme such that perturbative calculations are valid and simple.
According to the Appelquist–Carazzone theorem^{34}, heavy particles decouple automatically in a massdependent scheme, and their impact on the effective theory will be inversely proportional to the heavy particle mass M and disappear in the limit of an infinitely large mass. Nevertheless, higherorder calculations in this scheme become quite involved. The massindependent schemes, such as the modified minimal subtraction scheme () (refs 35, 36, 37), have been suggested for practical computations in effective theories^{38}, where the strategy to construct a selfconsistent effective theory in the scheme is outlined and applied to the determination of the heavy gauge boson mass M_{G} in the SU(5) GUT^{39}.
One problem for the massindependent scheme is that the heavy particles contribute equally to the socalled beta functions for gauge coupling constants, leading to an incorrect evolution at a lowenergy scale . The solution to this problem is to decouple heavy particles by hand and match the effective theory with the full theory at μ=M_{G} so that the same physical results can be produced in the effective theory as in the full theory. At any other energy scale below M_{G}, gauge coupling constants are governed by renormalization group running, which will be discussed in the following subsection. Therefore, if there are several heavy particles with very different masses, we should decouple them one by one to obtain a series of effective theories. The matching conditions (that is, the boundary conditions) at each mass scale are crucial for the effective theory to work below this scale. As a consequence, physical quantities (such as coupling constants and masses) may dramatically change at a decoupling scale or mass threshold. To figure out threshold effects, one has first to start with the full theory and construct the effective theories following the above strategy.
Renormalization group running
The renormalization group was invented in 1953 by Stückelberg and Petermann^{40}. However, it was GellMann and Low^{41} who studied the shortdistance behaviour of the photon propagator in quantum electrodynamics in 1954 by using the renormalization group approach. The important role played by the renormalization group in GellMann and Low’s work was clarified in 1956 by Bogoliubov and Shirkov^{42}. The same approach was applied by Wilson to study critical phenomena and explain how phase transitions take place^{43,44,45}.
The essential idea of the renormalization group stems from the fact that the theory is invariant under the change of renormalization prescription. More explicitly, if the theory is renormalized at a mass scale μ, any change of μ will be compensated by changes in the renormalized coupling constant g(μ) and the mass m(μ) such that the theory remains the same. By requiring that the physical quantities, for example, the Smatrix element S[μ,g(μ),m(μ)], are invariant under this transformation, namely μdS/dμ=0, one can derive
which is a specific form of the Callan–Symanzik equation^{46,47}. Note that we have introduced the RGEs for the coupling constant and the mass
where β(g) and γ_{m}(g) are the beta function and the anomalous dimension, respectively, depending only on the coupling constant g in the scheme.
As pointed out by Weinberg^{48} a long time ago, the standard electroweak model can be regarded as an effective theory at low energies, and the impact of new physics at highenergy scales can be described by higherdimensional operators, which are composed of the already known SM fields. If the SM gauge symmetry is preserved, but the accidental symmetry of lepton number is violated, there will be a unique dimensionfive operator , where and H stand for the SM lepton and Higgs doublets, respectively. After spontaneous breakdown of electroweak gauge symmetry, neutrinos acquire finite masses from the socalled Weinberg operator . Therefore, neutrinos are assumed to be Majorana particles in this case. It is expected that the lightness of neutrinos can be ascribed to the existence of a superhighenergy scale. Now, it becomes clear that if neutrino masses originate from some dynamics at a highenergy scale, such as the GUT scale, neutrino parameters including leptonic mixing parameters and neutrino masses will evolve according to their RGEs as the energy scale goes down to where the parameters are actually measured in lowenergy experiments.
Neutrino mass models
To generate tiny neutrino masses, one has to go beyond the SM and extend its particle content, or its symmetry structure, or both. In this section, we summarize several typical neutrino mass models, which are natural extensions of the SM that have attracted a lot of attention in the past decades. In Fig. 1, the Feynman diagrams for neutrino mass generation in those models are shown.
Canonical seesaw models
As the Higgs particle has recently been discovered in the ATLAS^{21} and CMS^{22} experiments at the Large Hadron Collider, the SM gauge symmetry SU(2)_{L} × U(1)_{Y} and its spontaneous breaking via the Higgs mechanism seem to work perfectly in describing the electromagnetic and weak interactions. On the other hand, nonzero neutrino masses indicate that the SM may just be an effective theory below and around the electroweak scale Λ_{EW}=10^{2} GeV. Thus, one can preserve the SM gauge symmetry structure and take into account all higherdimensional operators, which are relevant for neutrino masses, as pointed out by Weinberg^{48}. The total Lagrangian is
where _{SM} denotes the SM Lagrangian, and H stand for the SM lepton and Higgs doublets, respectively. The coefficients κ_{αβ} (α,β=e,μ,τ) are of mass dimension −1 and related to the Majorana neutrino mass matrix as M_{ν}=κ‹H›^{2}, where ‹H›≈174 GeV is the vacuum expectation value of H.
One of the simplest extensions of the SM, leading to the Weinberg operator, is the socalled typeI seesaw model, in which three righthanded singlet neutrinos ν_{R} are introduced. Since the ν_{R}’s are neutral under transformations of the SM gauge symmetry, they can have Majorana mass terms, namely, their masses are the eigenvalues of a complex and symmetric mass matrix M_{R}. On the other hand, they are coupled to the lepton and Higgs doublets via a Yukawatype interaction with a coupling matrix Y_{ν}. Since the masses of righthanded neutrinos are not subject to electroweak symmetry breaking, we can assume that and integrate out the three ν_{R}’s. At a lowerenergy scale, one obtains the Weinberg operator with . Therefore, the smallness of neutrino masses can be attributed to the heaviness of the ν_{R}’s (refs 13, 14, 15, 16, 17).
In the typeII seesaw model^{49,50,51,52,53,54}, the scalar sector of the SM is enlarged with a Higgs triplet Δ. To avoid an unwanted Goldstone boson associated with the spontaneous breakdown of the global U(1) lepton number symmetry, one can couple the Higgs triplet to the lepton doublet with a Yukawa coupling matrix Y_{Δ}, and simultaneously to the Higgs doublet with a mass parameter μ_{Δ}. Assuming that the Higgs triplet mass M_{Δ} is well above the electroweak scale, that is, , we can integrate out Δ to obtain the Weinberg operator with κ=Y_{Δ}μ_{Δ}/M_{Δ}^{2}, indicating that the neutrino masses are suppressed by M_{Δ}.
In the typeIII seesaw model^{55}, one introduces three fermion triplets Σ_{i} (i=1,2,3) and couple them to the lepton and Higgs doublets with a Yukawa coupling matrix Y_{Σ}. In each Σ_{i}, there are three heavy fermions: two charged fermions and one neutral fermion . Given a Majorana mass matrix M_{Σ} of the fermion triplets and , we can construct an effective theory without the heavy Σ_{i}’s at a lowerenergy scale. In this effective theory, the same Weinberg operator for neutrino masses can be obtained and the coefficient is identified as . One can observe that the ’s are playing the same role in generating neutrino masses as the ν_{R}’s in the typeI seesaw model. However, due to their gauge interaction, the fermion triplets are subject to more restrictive constraints from leptonflavourviolating decays of charged leptons and direct collider searches.
A common feature of the above three seesaw models is the existence of superheavy particles. Given neutrino masses and ‹H›~100 GeV, one can estimate the seesaw scale Λ_{SS}~10^{14} GeV. Therefore, an effective theory with the same Weinberg operator is justified at any scale between Λ_{EW} and Λ_{SS}. Although the leptogenesis mechanism for the matter–antimatter asymmetry can be perfectly implemented in the seesaw framework, the heaviness of new particles renders the seesaw models difficult to be tested in lowenergy and collider experiments.
Inverse seesaw model
To lower the typical seesaw scale Λ_{SS} in a natural way, one can extend the typeI seesaw model by adding three righthanded singlet fermions S_{R} and one Higgs singlet Φ, both of which are coupled to the ν_{R}’s by a Yukawa coupling matrix Y_{S}. A proper assignment of quantum numbers under a specific global symmetry can be used to forbid the Majorana mass term of ν_{R} and the ν_{R}–Φ Yukawa interaction. However, the mixing between ν_{R} and S_{R} is allowed through a Dirac mass term M_{S}=Y_{S}‹Φ›, so is the Majorana mass term . In this setup, the Majorana mass matrix for three light neutrinos is given by , where M_{D}≡Y_{ν}‹H› is the Dirac neutrino mass matrix as in the typeI seesaw model.
Given and , the subeV neutrino masses can be achieved by assuming μ_{S}~1 keV. In this inverse seesaw model^{56}, the neutrino masses are not only suppressed by the ratio of the electroweak and seesaw energy scales, that is, Λ_{EW}/Λ_{SS}=M_{D}/M_{S}~10^{−2}, but also by the tiny lepton numberviolating mass parameter μ_{S} compared with the ordinary seesaw scale. The smallness of μ_{S} is natural in the sense that the model preserves the lepton number symmetry in the limit μ_{S}→0 (ref. 57). In contrast to the ordinary seesaw models, the inverse seesaw model is testable through nonunitarity effects in neutrino oscillation experiments^{58}, leptonflavourviolating decays of charged leptons^{59,60,61} and collider experiments^{62,63}.
Scotogenic model
A radiative mechanism for neutrino mass generation is to attribute the smallness of neutrino masses to loop suppression instead of the existence of superheavy particles^{64,65,66,67,68}. One interesting model of this type is the socalled scotogenic model^{67}, where three ν_{R}’s and one extra Higgs doublet η are added to the SM. Furthermore, a Z_{2} symmetry is imposed on the model such that all SM fields are even, while ν_{R} and η are odd. Even though the SU(2)_{L} × U(1)_{Y} quantum numbers of η are the same as the SM Higgs doublet and the ν_{R}’s have a Majorana mass term, the Dirac neutrino mass term is forbidden by the Z_{2} symmetry and neutrino masses are vanishing at tree level.
In the scotogenic model, neutrino masses appear first at oneloop level and the exact Z_{2} symmetry guarantees the stability of one neutral scalar boson (from the Higgs doublet η), which would be a good candidate for a darkmatter particle^{67}. Due to loop suppression, subeV neutrino masses can be obtained even when ν_{R}’s and scalar particles are at the TeV scale. Therefore, this model has observable effects in leptonflavourviolating processes, relic density of dark matter and collider phenomenology^{67}.
Dirac neutrino model
Finally, we consider the Dirac neutrino model. In the SM model, both quarks and charged leptons acquire their masses through Yukawa interactions with the Higgs doublet. After introducing three ν_{R}’s, one can do exactly the same thing for neutrinos, and thus, tiny neutrino masses can be ascribed to the smallness of neutrino Yukawa couplings. One difficulty with the Dirac neutrino model is why the fermion masses span 12 orders of magnitude, exaggerating the strong hierarchy problem of fermion masses in the SM. Solutions to the above problem can be found in extradimensional models^{69}, where the SM particles are confined to a threedimensional brane and the ν_{R}’s are allowed to feel one or more extra dimensions^{70}. In this case, the neutrino Yukawa couplings are highly suppressed by the large volume of the extra dimensions. Another solution is to implement a radiative mechanism, as in the scotogenic model, such that light neutrino masses are due to loop suppression^{71,72,73,74}. See Fig. 1 for an illustration. However, in both kinds of models, an additional U(1) symmetry (that is, lepton number conservation) has to be enforced to forbid a Majorana mass term.
Running behaviour of neutrino parameters
Now, we proceed to discuss the running behaviour of neutrino parameters. First of all, we note that there are two different ways to study the renormalization group running. In the top–down scenario, a full theory is known at the highenergy scale and the theoretical predictions for neutrino parameters are given as initial conditions. At the threshold of heavy particle decoupling, one has to match the resulting effective theory with the full theory, so that the unknown parameters in the effective theory can be determined and used to reproduce the same physical results as in the full theory. Then, the running is continued in the effective theory. This procedure should be repeated in the case of multiple particle thresholds until a lowenergy scale where the neutrino parameters are measured. In the bottom–up scenario, we start with the experimental values of neutrino parameters at a lowenergy scale, and evolve them by using the RGEs in the effective theory to the first particle threshold. At this moment, more input or assumptions about the dynamics above the threshold are needed for the running to continue. Otherwise, the running is terminated and some useful information on the full theory cannot be obtained.
In the following, we will focus on the bottom–up approach and explore the implications of measurements of neutrino parameters for the dynamics at a highenergy scale, where a full theory of neutrino masses and lepton flavour mixing may exist. However, we shall also comment on the threshold effects in the top–down scenario once a specific flavour model is assumed. In the effective theory, where the SM is extended by the Weinberg operator, the RGE for the effective neutrino mass parameter κ was first derived in refs 75, 76, and revised in ref. 77. In general, we have the RGE for κ given by
where t=ln(μ/Λ_{EW}) and Y_{l} stands for the chargedlepton Yukawa coupling matrix. In equation (5), C_{κ} is a constant, while α_{κ} depends on the gauge couplings and all Yukawa coupling matrices of the charged fermions. Given the initial values of all relevant coupling constants and masses at Λ_{EW}, one can evaluate the neutrino parameters at any energy scale between Λ_{EW} and a cutoff scale Λ, after solving equation (5) together with the RGEs of the other model parameters and diagonalizing κ. Since κ is diagonalized by the lepton flavour mixing matrix U(θ_{12},θ_{13},θ_{23},δ,ρ,σ) in the basis where Y_{l} is diagonal, one can derive, using equation (5), the individual RGEs for the leptonic mixing angles {θ_{12},θ_{13},θ_{23}}, the CPviolating phases {δ,ρ,σ}, and the neutrino mass eigenvalues {m_{1},m_{2},m_{3}}, which can be found in refs 78, 79, 80.
The Standard Model
In the framework of the SM, the relevant coefficients in equation (5) are given by and , where only the Yukawa couplings of the heaviest charged lepton and quark are retained, and λ is the quartic Higgs selfcoupling constant. Since the Yukawa couplings of charged leptons are small compared with gauge couplings, the evolution of neutrino masses can be essentially described by a common scaling factor. For the running of the leptonic mixing angles, the contribution from tau Yukawa coupling y_{τ}=m_{τ}/‹H›~0.01 is dominant. However, y_{τ} itself is already a very small number, so one expects that the running effects of all three leptonic mixing angles are generally insignificant.
On the other hand, the evolution of the leptonic mixing angles can be enhanced if the neutrino mass spectrum is quasidegenerate, that is, . In particular, the leptonic mixing angle θ_{12} has the strongest running effects, partly due to . In the limit of quasidegenerate mass spectrum and CP conservation, the RGEs for the two neutrino masssquared differences and the three leptonic mixing angles are given by
where the relevant coefficients are presented in Table 2. It is now straightforward to observe that the evolution of θ_{12} is enhanced by a factor of , compared with that of θ_{13} and θ_{23}. For illustration, the evolution of θ_{12} from M_{Z}=91.19 GeV to Λ=10^{10} GeV is shown in Fig. 2. At M_{Z}, the gauge coupling constants and the quark mixing parameters are taken from the Particle Data Group^{3}, the quark and chargedlepton masses from refs 81, 82 and the leptonic mixing parameters are set to the bestfit values from the NuFit group^{30}. The Higgs mass M_{H}=126 GeV is assumed to be consistent with the latest measurements by the ATLAS^{21} and CMS^{22} experiments. It is worthwhile to mention that M_{H} or equivalently the Higgs selfcoupling constant affects the running of neutrino masses, and also the SM vacuum stability^{23}. Finally, a quasidegenerate neutrino mass spectrum is adopted with the lightest neutrino mass m_{1}=0.2 eV and the Majorana CPviolating phases {ρ,σ} are set to zero. Even with these extremely optimistic assumptions, the value of θ_{12} turns out to be only larger by 1° at Λ=10^{10} GeV than at M_{Z}.
The previous observations apply well to seesaw models with Λ=10^{10} GeV identified as the mass of the lightest new particle. Above the seesaw threshold, the running of neutrino parameters has also been studied in the complete typeI^{83,84,85}, typeII^{86,87,88} and typeIII^{89} seesaw models. However, for lowscale neutrino mass models, there exist new particles at the TeV scale. Therefore, the running behaviour of neutrino parameters can be significantly changed by threshold effects in the inverse seesaw model^{90,91} and the scotogenic model^{92}. In the Dirac neutrino model, the RGEs of the neutrino parameters have also been derived and investigated in detail^{93}.
Supersymmetric models
In the minimal supersymmetric extension of the SM (MSSM), all fermions have bosonic partners, and vice versa^{94}. Although there is so far no direct hint on supersymmetry, the MSSM is regarded as one of the most natural alternatives to the SM for its three salient features: (1) elimination of the finetuning or hierarchy problem; (2) implication for grand unification of gauge coupling constants; (3) candidates for the dark matter. Hence, neutrino mass models in the supersymmetric framework are extensively studied in the literature^{78}.
In the MSSM extended with the Weinberg operator, the corresponding coefficients in equation (5) are and . The neutrino mass matrix is then given by with tan β being the ratio of the vacuum expectation values of the two Higgs doublets in the MSSM. Similar to the SM, the running of the leptonic mixing angles is dominated by the tau Yukawa coupling . However, now y_{τ} can be remarkably larger than its value in the SM if a large value of tan β is chosen. Consequently, apart from the enhancement due to a quasidegenerate neutrino mass spectrum, the running effects of the leptonic mixing angles can be enlarged by tan β. In Fig. 2, we show the evolution of θ_{12} in the MSSM with tan β=10, where the input values at M_{Z} are the same as in the SM. In addition, the supersymmetry breaking scale is assumed to be 1 TeV, below which the SM works well as an effective theory. The value of θ_{12} decreases with respect to an increasing energy scale, whereas it increases in the SM. This is due to the opposite signs of and .
As an example for the top–down approach, one considers a bimaximalmixing pattern (that is, θ_{12}=θ_{23}=45° and θ_{13}=0) at the GUT scale Λ_{GUT}=2 × 10^{16} GeV (refs 84, 95). It is worthwhile to mention that the leptonic mixing angles above the seesaw scale arise from the diagonalization of , and the leptonic mixing angles and the neutrino masses at this scale can be viewed as a convenient parametrization of , which is a combination of fundamental model parameters Y_{ν} and M_{R}. Therefore, a bimaximalmixing pattern may result from a flavour symmetry at the GUT scale. For a complete typeI seesaw model at Λ_{GUT}, the full flavour structure of the neutrino Yukawa coupling matrix Y_{ν} should be specified, and the mass matrix of righthanded neutrinos is reconstructed from the light neutrino mass matrix and Y_{ν} from the seesaw formula. See ref. 84 for the other input parameters. In Fig. 3, the running behaviour of the three leptonic mixing angles are depicted, where the grayshaded areas stand for the decoupling of three righthanded neutrinos at M_{3}=8.1 × 10^{13} GeV, M_{2}=2.1 × 10^{10} GeV and M_{1}=5.5 × 10^{8} GeV. As one can observe from Fig. 3, the decoupling of the heaviest righthanded neutrino and the matching between the first effective theory and the full theory have remarkable impact on the running of θ_{12} and θ_{13}. This impact depends on the presumed flavour structure in the lepton sector, indicating that the running of neutrino parameters has to be taken into account in the flavour model at a superhighenergy scale.
In the MSSM, it is in general expected that running effects of neutrino parameters are significant, in particular for large values of tan β and a quasidegenerate neutrino mass spectrum. This generic feature should also be applicable to supersymmetric versions of neutrino mass models discussed in the previous section.
Extradimensional models
The existence of one or more extra spatial dimensions was first considered by Kaluza^{96} and Klein^{97} in the 1920s. The recent interest in extra dimensions and their implications for particle physics was revived by the seminal works in refs 98, 99, 100. In extradimensional models, the fundamental energy scale for gravity can be as low as a few TeV, solving the gauge hierarchy problem of the SM. Furthermore, the excited Kaluza–Klein (KK) modes of the SM fields serve as promising candidates for cold dark matter. See ref. 101, for a brief review.
As an interesting example for the running of neutrino parameters in extradimensional models, we consider the socalled universal extradimensional model (UEDM) first introduced in ref. 102, in which all SM fields are allowed to propagate in one or more compact extra dimensions. Since the KK number is conserved and the excited KK modes manifest themselves only at loop level, current mass bound on the first KK excitation from electroweak precision measurements and direct collider searches is just about a few hundred GeV (ref. 102). In the fivedimensional UEDM, the corresponding coefficients in equation (5) are and , where s=⌊μ/μ_{0}⌋ is the number of excited KK modes at the energy scale μ. Note that μ_{0} denotes the mass of the first KK excitation, or equivalently R=μ_{0}^{−1} is the radius of the compact extra dimension. In contrast to the SM and the MSSM, the running of κ in the UEDM obeys a power law due to the increasing number of excited KK modes, implying a significant boost in the running^{103,104,105}. The reason is simply that, at a given energy scale μ, we have an effective theory with s=⌊μ/μ_{0}⌋ new particles, which will run in the loops and contribute to the RGEs of neutrino parameters.
In Fig. 2, the evolution of θ_{12} in the fivedimensional UEDM is shown, where the input parameters at M_{Z} are the same as in the SM and a cutoff scale Λ=40 TeV has been chosen to guarantee that a perturbative effective theory is valid. One can observe that the running effect is significant even in such a narrow energy range. It is worthwhile to mention that θ_{12} increases with respect to an increasing energy scale in both the SM and the UEDM, whereas it decreases in the MSSM.
Generic features
Now, we summarize the generic features of the running of neutrino parameters in the SM, the MSSM and the UEDM. First, due to small Yukawa couplings of charged leptons in the SM, the evolution of the leptonic mixing angles is insignificant, even in the case of a quasidegenerate neutrino mass spectrum. The running effects can be remarkably enhanced in the MSSM through a relatively large value of tan β, and instead through the number of excited KK modes in the UEDM. Second, among the three leptonic mixing angles, θ_{12} has the strongest running effect due to an enhancement factor . The running of θ_{12} in the SM and the UEDM is in the opposite direction to that in the MSSM. However, the actual running behaviour also crucially depends on the choice of the currently unconstrained leptonic CPviolating phases^{78}. The running neutrino masses at highenergy scales can be approximately obtained by multiplying a common scaling factor, depending on the evolution of the gauge couplings. Third, the running effects of the leptonic CPviolating phases have been studied in detail in refs 78, 106, 107, 108, where the evolution of the three CPviolating phases has been found to be entangled. Consequently, a nonzero Dirac CPviolating phase can be radiatively generated even if it is assumed to be zero at a highenergy scale, and vice versa.
Finally, it is worth mentioning that threshold effects may significantly change the running behaviour of different neutrino parameters. However, the accurate description of threshold effects is only possible if the full theory is exactly known.
Phenomenological implications
The running of neutrino parameters has important implications for flavour model building, the matter–antimatter asymmetry via the leptogenesis mechanism, and the extradimensional models. We now sketch the essential points and refer interested readers to relevant references.
Flavour model building
In connection with flavour mixing in the quark sector, flavour models are usually built at a highenergy scale, for example, the GUT scale. As for flavour model building, the running effects should be taken into account in general, and for the case of quasidegenerate neutrino masses in particular. The running effects of mixing parameters can be used to interpret the discrepancy between quark and lepton flavour mixing^{109,110}. As a possible symmetry between quarks and leptons, quark–lepton complementarity relations, such as and , where the superscripts specify the mixing angles in the quark and lepton sectors, have been conjectured^{111,112}. Radiative corrections to these relations have been calculated in the typeI seesaw model^{113}.
To describe the observed lepton mixing pattern, one may impose a discrete flavour symmetry on the generic Lagrangian^{19,20}. As discussed in the previous section, a bimaximalmixing pattern (that is, and θ_{13}=0) at Λ_{GUT} turns out to be compatible with current neutrino oscillation data if running effects are taken into account^{95}. In addition to bimaximal mixing^{114}, tribimaximal^{115,116,117}, democratic^{118} and tetramaximal mixing^{119} patterns have been proposed to describe lepton flavour mixing, and their radiative corrections have also been examined^{120,121,122,123,124,125,126}.
Matter–antimatter asymmetry
It remains an unanswered question why our visible world is made of matter rather than antimatter. From cosmological observations, the ratio between baryon number density and photon number density η_{b}=(6.19±0.15) × 10^{−10} has been precisely determined^{3}. One of the most attractive mechanisms for a dynamic generation of baryon asymmetry is leptogenesis^{18}, which works perfectly in various seesaw models for neutrino mass generation.
Take the typeI seesaw model, for example, where three heavy righthanded neutrinos are introduced. In the early Universe, when the temperature is as high as the masses of heavy neutrinos, they can be thermally produced and decay into the SM particles, mainly lepton and Higgs doublets. If the neutrino Yukawa couplings are complex, heavy neutrinos decay into leptons and antileptons in different ways. When the Universe cools down, CPviolating decays go out of thermal equilibrium and a lepton asymmetry can be generated, which will be further converted into a baryon asymmetry.
The final baryon asymmetry η_{b}≈0.96 × 10^{−2}ε_{1}κ_{f} depends on the CP asymmetry ε_{1} from the decays of the lightest heavy neutrino, and the efficiency factor κ_{f} from the solution to a set of Boltzmann equations^{127}. Moreover, the maximal value of ε_{1} can be derived
where m denotes the mass of heaviest ordinary neutrino^{128}. Now, it is evident that the running of neutrino masses from the lowenergy scale to M_{1} (that is, the mass of ν_{1R}) should be taken into account^{84,129}. As the evolution of neutrino masses can be described by a common scaling factor and they become larger at a higherenergy scale, the maximum of the CP asymmetry scales upwards as neutrino masses. However, larger values of neutrino masses at M_{1} imply larger Yukawa couplings, which enhance the washout of the lepton asymmetry, and thus reduce κ_{f}. The outcome from the competition between the enhancement of ε_{1} and the reduction of κ_{f} depends on the neutrino mass spectrum, and also on the value of tan β in the MSSM^{78}. See ref. 130, for a review on the recent development of leptogenesis in seesaw models.
Bounds on extra dimensions
A general feature of quantum field theories with extra spatial dimensions is that they are nonrenormalizable^{131}, since there exist infinite towers of KK states appearing in the loops of quantum processes. As pointed out in ref. 131, the higherdimensional theories could preserve renormalizability if they are truncated at a certain energy scale Λ (see Fig. 2), below which only a finite number of KK modes is present. In the UEDM, Λ is usually taken to be the energy scale where the gauge couplings become nonperturbative^{132}, but it could also be related to a unification scale for the gauge couplings^{131}.
The recent discovery of a Higgs particle with M_{H}=126 GeV leads to a reconsideration of the stability of the SM vacuum^{82,23}. The instability is essentially induced by the fact that the Higgs selfcoupling constant λ runs into a negative value at a highenergy scale. Since the model parameters have a powerlaw running in the UEDM, in contrast to a logarithmic running in the ordinary fourdimensional theories, the requirement of vacuum stability will place a restrictive bound on the cutoff scale Λ and the radius of extra dimensions R. It has been found that ΛR<5 for R^{−1}=1 TeV in the fivedimensional UEDM^{133}, while this bound becomes more stringent ΛR<2.5 in the sixdimensional UEDM^{104}, which can be translated into the maximal number of KK modes being five and two, respectively. As a consequence, the running of neutrino parameters in these models will be limited to a narrow energy range.
Outlook
Our knowledge about neutrinos has been greatly extended in the past decades, especially due to a number of elegant neutrino oscillation experiments. As for the leptonic mixing parameters, we are entering into the era of precision measurements of three leptonic mixing angles and two neutrino masssquared differences. The determination of the neutrino mass hierarchy and the discovery of leptonic CP violation are now the primary goals of the ongoing and upcoming neutrino oscillation experiments. On the other hand, the tritium beta decay and neutrinoless doublebeta decay experiments, together with cosmological observations, will probe the absolute scale of neutrino masses. Whether or not neutrinos are their own antiparticles will also be clarified if neutrinoless doublebeta decay is observed. Therefore, we will obtain more information about the neutrino parameters at the lowenergy scale.
However, the origin of neutrino masses and lepton flavour mixing remains a big puzzle in particle physics. In this review article, we have elaborated on the evolution of neutrino parameters from the lowenergy scale to a superhighenergy scale, where new physics may appear and take the responsibility for generating neutrino masses. The running effects of neutrino parameters can be very significant and should be taken into account in searching for a true theory of neutrino masses and lepton flavour mixing. On the other hand, the successful applications of renormalization group running in neutrino physics, and more generally in elementary particle physics and condensed matter physics, will demonstrate the deep connection between different branches of physical sciences and the amazing power of quantum field theories in describing nature.
In the foreseeable future, with direct searches at colliders and precision measurements of quark and lepton flavour mixing parameters, we hope that all observations will finally converge into hints for new physics beyond the SM and lead us to a complete theory of fermion masses, flavour mixing and CP violation.
Additional information
How to cite this article: Ohlsson, T. and Zhou, S. Renormalization group running of neutrino parameters. Nat. Commun. 5:5153 doi: 10.1038/ncomms6153 (2014).
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Acknowledgements
We would like to thank Mattias Blennow and He Zhang for useful discussions and comments. This work was supported by the Swedish Research Council (Vetenskapsrådet), contract no. 62120113985.
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Ohlsson, T., Zhou, S. Renormalization group running of neutrino parameters. Nat Commun 5, 5153 (2014). https://doi.org/10.1038/ncomms6153
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DOI: https://doi.org/10.1038/ncomms6153
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