Generalized Tri-bimaximal Neutrino Mixing and Its Sensitivity to Radiative Corrections

We argue that the tri-bimaximal neutrino mixing pattern $V_0$ or its generalized form $V'_0$, which includes two arbitrary Majorana phases of CP violation, may result from an underlying flavor symmetry at a superhigh energy scale close to the seesaw scale ($\sim 10^{14}$ GeV). Taking the working assumption that three neutrino masses are nearly degenerate, we calculate radiative corrections to $V_0$ and $V'_0$ in their evolution down to the electroweak scale ($\sim 10^2$ GeV). Three mixing angles of $V_0$ or $V'_0$ are essentially stable against radiative corrections in the standard model (SM). In the minimal supersymmetric standard model (MSSM), however, $V_0$ is in general disfavored and $V'_0$ can be compatible with current neutrino oscillation data if its two Majorana phases $\alpha^{}_1$ and $\alpha^{}_2$ are properly fine-tuned. We also find that it is possible to radiatively generate the CP-violating phase $\delta$ from $\alpha^{}_1$ and $\alpha^{}_2$, and \delta may keep on staying at its fixed point in either the SM or the MSSM.


Abstract
We argue that the tri-bimaximal neutrino mixing pattern V 0 or its generalized form V ′ 0 , which includes two arbitrary Majorana phases of CP violation, may result from an underlying flavor symmetry at a superhigh energy scale close to the seesaw scale (∼ 10 14 GeV). Taking the working assumption that three neutrino masses are nearly degenerate, we calculate radiative corrections to V 0 and V ′ 0 in their evolution down to the electroweak scale (∼ 10 2 GeV). Three mixing angles of V 0 or V ′ 0 are essentially stable against radiative corrections in the standard model (SM). In the minimal supersymmetric standard model (MSSM), however, V 0 is in general disfavored and V ′ 0 can be compatible with current neutrino oscillation data if its two Majorana phases α 1 and α 2 are properly fine-tuned. We also find that it is possible to radiatively generate the CP-violating phase δ from α 1 and α 2 , and δ may keep on staying at its quasi-fixed point in either the SM or the MSSM. PACS number(s): 14.60.Pq,13.10.+q,25.30.Pt Typeset using REVT E X * Mailing address 1 Current solar [1], atmospheric [2], reactor [3] and accelerator [4] neutrino experiments have provided us with very convincing evidence for the existence of neutrino oscillations, a quantum phenomenon which can naturally occur if neutrinos are massive and lepton flavors are mixed. The property of lepton flavor mixing can be described by a 3 × 3 unitary matrix V . A parametrization of V , advocated by the Particle Data Group [5], reads as where c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 23 and 13). The phase parameters α 1 and α 2 are usually referred as to the Majorana CP-violating phases, because they have nothing to do with CP or T violation in the neutrino-neutrino and antineutrino-antineutrino oscillations. A global analysis of the present experimental data yields [6] 30 • ≤ θ 12 ≤ 38 • , 36 • ≤ θ 23 ≤ 54 • and 0 • ≤ θ 13 ≤ 10 • as well as ∆m 2 21 ≡ m 2 2 −m 2 1 = (7.2 · · · 8.9) ×10 −5 eV 2 and ∆m 2 32 ≡ m 2 3 − m 2 2 = ±(1.7 · · · 3.3) × 10 −3 eV 2 at the 99% confidence level. In contrast, three phases of V are entirely unrestricted. A variety of new neutrino experiments are underway, not only to detect the smallest flavor mixing angle θ 13 and the phase parameter δ, but also to constrain the Majorana phases α 1 and α 2 .
To interpret the observed neutrino mass spectrum and the observed bi-large neutrino mixing pattern, many theoretical and phenomenological models have been proposed and discussed [7]. A category of models or ansätze have attracted some particular attention, because they can give rise to the so-called tri-bimaximal neutrino mixing pattern [8]: Comparing between Eqs. (1) and (2), one may immediately observe that V 0 has θ 12 ≈ 35.3 • , θ 23 = 45 • , θ 13 = 0 • and α 1 = α 2 = 0 • . The phase parameter δ is not well-defined in V 0 , as a consequence of θ 13 = 0 • . The results sin 2 2θ 12 = 8/9 and sin 2 2θ 23 = 1 are in good agreement with current data on solar and atmospheric neutrino oscillations. It is straightforward to generalize V 0 in order to include two arbitrary Majorana phases, Although V 0 and V ′ 0 have the same impact on neutrino oscillations, their consequences on the neutrinoless double-beta decay are certainly different. In this sense, we refer to V ′ 0 as the generalized tri-bimaximal neutrino mixing pattern.
Such a special neutrino mixing pattern is in general expected to result from an underlying flavor symmetry (e.g., the discrete non-Abelian symmetry A 4 [9][10][11][12]) and its spontaneous or explicit breaking. The latter is always necessary, because a flavor symmetry itself cannot reproduce the observed lepton mass spectra and predict the realistic lepton mixing pattern simultaneously [13]. Specific and compelling constructions of this kind of flavor symmetry breaking are a real challenge and have been lacking, although some attempts have been made in the literature [7]. The energy scale, at which a proper flavor symmetry can be realized, may be considerably higher than the electroweak scale (Λ EW ∼ 10 2 GeV). This new physics (NP) scale Λ NP has actually been identified with other known scales in some model-building works [7], including the grand-unification-theory scale (Λ GUT ∼ 10 16 GeV) or the seesaw scale (Λ SS ∼ 10 14 GeV). In this case, radiative corrections to the relevant model parameters between Λ EW and Λ NP must be taken into account [14].
One may then ask whether the generalized tri-bimaximal neutrino mixing pattern is stable or not against radiative corrections, if it is derived from an underlying (broken) flavor symmetry within an unspecified mechanism at Λ NP (≫ Λ EW ). The main purpose of this paper is just to answer this question by considering both the standard model (SM) and its minimal supersymmetric extension (MSSM) below Λ NP . The only effective dimension-5 operator of light Majorana neutrinos reads as where H denotes the SM Higgs (or the MSSM Higgs with the appropriate hypercharge), L i (for i = 1, 2, 3) stand for the leptonic SU(2) L doublets, and κ is a symmetric neutrino coupling matrix. After spontaneous gauge symmetry breaking at Λ EW , we arrive at the effective neutrino mass matrix GeV and tan β is the ratio of the vacuum expectation values of two Higgs fields in the MSSM. Between Λ EW and Λ NP , the most important radiative correction to κ is proportional to ln(Λ NP /Λ EW ) and can be evaluated by using the one-loop renormalization group equations (RGEs) [14]. It is then possible to calculate the RGE effects on the lepton flavor mixing parameters analytically and numerically.
In the working assumption that three neutrino masses are nearly degenerate, we are going to calculate radiative corrections to V 0 and V ′ 0 . We show that both V 0 and V ′ 0 are stable against radiative corrections in the SM, but only V ′ 0 with the proper fine-tuning of (α 2 − α 1 ) is allowed in the MSSM. In addition, the CP-violating parameter δ can be radiatively generated from α 1 and α 2 . A peculiar feature of δ is that it may keep on staying at its quasi-fixed point in both the SM and the MSSM.
To be specific, we assume the masses of three Majorana neutrinos are nearly degenerate; i.e., m 1 ≈ m 2 ≈ m 3 . Such a working assumption makes sense at least for two practical reasons: (1) it might hint at the slight breaking of an exact S(3) permutation symmetry [20] or other possible flavor symmetries, from which the tri-bimaximal neutrino mixing pattern can naturally arise; and (2) more significant RGE running effects on three mixing angles and three CP-violating phases can manifest themselves in this interesting case. Furthermore, it is helpful to make some analytical approximations for the results obtained in Refs. [16][17][18][19] by taking account of the smallness of sin θ 13 and ∆m 2 21 /∆m 2 32 . We arrive at to a good degree of accuracy, where y τ denotes the tau-lepton Yukawa coupling. Given the approximate degeneracy of three neutrino masses, the RGEs of (θ 12 , θ 23 , θ 13 ) and (δ, α 1 , α 2 ) in Refs. [16][17][18][19] are simplified to as well as where Note that the χ-term is not negligible only in the special case that α 1 ≈ α 2 holds and s 13 is extremely small. Some qualitative comments on Eqs. (7) and (8) are in order.
(a) The mixing angle θ 12 is in general more sensitive to radiative corrections than θ 23 and θ 13 [16,18]. Given θ 12 ≈ 35.3 • as a result of the tri-bimaximal neutrino mixing at Λ NP , the RGE running effect has to be sufficiently suppressed such that θ 12 can finally run into the experimentally-allowed range 30 • ≤ θ 12 ≤ 38 • at low energies. This requirement is certainly satisfied in the SM with m 1 ∼ O(0.1 eV), in which θ 12 slightly decreases in the RGE evolution from Λ NP to Λ EW . In the MSSM, however, θ 12 must evolve to a bigger value at Λ EW . Hence the fine-tuning of (α 2 − α 1 ) is necessary for large values of tan β, so as to keep the evolution effect of θ 12 insignificant [16,18,21]. One can see that two Majorana phases of V ′ 0 play a very non-trivial role in the calculation of radiative corrections. In other words, the tri-bimaximal neutrino mixing pattern V 0 and its generalized counterpart V ′ 0 are distinguishable in model building by taking into account their different RGE running behaviors.
(b) Different from θ 12 , the mixing angles θ 23 and θ 13 are expected to be less sensitive to radiative corrections. Hence θ 23 at Λ EW may slightly deviate from its initial value θ 23 = 45 • at Λ NP as a consequence of the RGE running. On the other hand, θ 13 can be radiatively generated, although its value at Λ EW must be rather small. Note that the tri-bimaximal neutrino mixing pattern V 0 keeps CP-conserving in the RGE evolution from Λ NP to Λ EW . As for the generalized tri-bimaximal neutrino mixing scenario V ′ 0 , it is possible to generate both the mixing angle θ 13 and the CP-violating phase δ radiatively 2 . The latter results from two non-trivial Majorana phases (or one of them) in the RGE of δ. This observation implies that the RGE-corrected V ′ 0 may give rise to a non-vanishing Jarlskog invariant [22] at Λ EW , leading to observable CP violation in neutrino oscillations.
(c) It should be noted that δ is not well-defined at Λ NP , where θ 13 is exactly vanishing in either V 0 or V ′ 0 . This point can clearly be seen from Eq. (8), in which the derivative of δ diverges in the θ 13 → 0 limit. Nevertheless, it has been shown in Ref. [18] that there exists an analytic continuation of δ, such that it remains well-defined even when θ 13 approaches zero. Hence θ 13 and δ can simultaneously be generated from the RGE running effects in the generalized tri-bimaximal neutrino mixing scenario. A peculiar RGE behavior of δ is that it can keep on staying at its quasi-fixed point just below Λ NP , as one can see in the subsequent numerical examples.
3 Now let us quantitatively illustrate radiative corrections to V 0 and V ′ 0 by taking a few numerical examples 3 . The eigenvalues of Y l at Λ NP are chosen in such a way that 2 For a more generic study of this problem, we refer readers to the works done by Casas et al in Ref. [16], Antusch et al in Ref. [18], and Luo et al in Ref. [19]. 3 Our numerical calculations follow a "running and diagonalizing" procedure [18]: we first compute the RGE evolution of lepton mass matrices and then extract their mass eigenvalues and flavor mixing parameters at Λ EW . Because θ 13 = 0 holds exactly at Λ NP and δ is always associated with s 13 in the chosen parametrization of V , any initial input of δ is allowed but it does not take any effect in the RGE running. The finite running result of δ is actually attributed to the initial values of two Majorana phases α 1 and α 2 . they can correctly run to their low-energy values. We typically take m 1 ∼ 0.2 eV, which is consistent with the working assumption m 1 ≈ m 2 ≈ m 3 made above. The initial values of three mixing angles at Λ NP are fixed: θ 12 ≈ 35.3 • , θ 23 = 45 • and θ 13 = 0 • , predicted by V 0 or V ′ 0 . Two unknown Majorana phases in V ′ 0 are adjustable in our numerical calculations, in which we choose m H = 140 GeV (SM) or tan β = 10 (MSSM) as a typical and instructive input. The primary results are shown in Tables I and II together with Figs. 1 and 2. Some discussions are in order.
(A) In the SM. It is demonstrated that the RGE running effects on three mixing angles (θ 12 , θ 23 and θ 13 ) are small enough in the SM, thus either V 0 or V ′ 0 at Λ NP can agree with current neutrino oscillation data at low energies. For V 0 , CP conservation keeps to hold in the RGE evolution from Λ NP to Λ EW (see Case I in Table I). But for V ′ 0 , the radiative generation of δ can always take place, only if α 1 or α 2 is initially non-vanishing (see Cases II and III in Table I). Because of θ 13 = 0 • at Λ NP , an extraordinarily large RGE correction to the CP-violating phase δ arises from the term proportional to 1/s 13 on the right-hand side of Eq. (8). It turns out that δ keeps on staying at its quasi-fixed point (see Fig. 1 for illustration). In contrast, the running of α 1 and α 2 is so tiny that they are essentially stable between Λ NP and Λ EW .
We find that the analytical approximation made in Eq. (8) together with Eq. (9) is helpful to understand the quasi-fixed point of the CP-violating phase δ in its RGE evolution from Λ NP to Λ EW . At such a quasi-fixed point, the condition dδ/dt ≈ 0 should be satisfied. This observation essentially implies that either sin δ +α 2 − sin δ +α 1 ≈ 0 (10) withα 1 =α 2 or sin δ +α 1 + sinδ ≈ 0 withα 1 ≈α 2 should hold in the leading-order approximation; i.e., the 1/s 13 term in the expression of dδ/dt must be strongly suppressed around the quasi-fixed point, where the values of three CP-violating phases are denoted byδ,α 1 andα 2 . The simple solutions to Eqs. (10) and (11) arê andδ (for n = 0, ±1, ±2, · · ·), respectively. These two possibilities correspond to the numerical examples given in Cases II and III in Table I or Fig. 1(a) and (b). The tiny magnitude of θ 13 at Λ EW implies that it is easy to rule out V 0 or V ′ 0 at Λ NP , after θ 13 = 0 • is experimentally established. Indeed, the sensitivity of a few currently-proposed reactor neutrino experiments to θ 13 is at the level of θ 13 ∼ 3 • or sin 2 2θ 13 ∼ 0.01 [23]. Since θ 13 is considerably suppressed, as shown in our numerical examples, it will be extremely difficult to measure δ (even if δ ∼ ±90 • ) in any long-baseline neutrino oscillation experiments. In this case, only the neutrinoless double-beta decay could be used to distinguish V ′ 0 from V 0 , because its effective mass m ee is sensitive to the Majorana phases 4 . We know that V 0 predicts m ee ≈ m 1 and V ′ 0 yields 14) in the assumption that three neutrino masses are nearly degenerate. The ratio of m ′ ee to m ee may take its minimal value [ m ′ ee / m ee ] min ≈ cos 2θ 12 ≈ 0.34 (for θ 12 ≈ 35 • ), when α 2 − α 1 ≈ ±π is satisfied.
We remark that the RGE running behaviors of V 0 and V ′ 0 are quite different in the SM. This difference can definitely affect the model building in understanding the origin of V 0 or V ′ 0 . It is worth mentioning that a particular mass model of charged leptons and neutrinos with the non-Abelian symmetry A 4 , from which V ′ 0 can be derived at a superhigh energy scale, has been proposed and discussed in Ref. [11]. Its low-energy consequences are actually within the scope of our generic RGE analysis.
(B) In the MSSM. We have pointed out that the mixing angle θ 12 is quite sensitive to radiative corrections, and it always runs to a bigger value at Λ EW from the initial value θ 12 ≈ 35.3 • at Λ NP in the MSSM. Considering the tri-bimaximal neutrino mixing pattern V 0 and taking tan β = 10 as a typical input, we find that the RGE running result of θ 12 at Λ EW exceeds its experimental upper bound (i.e., θ 12 ≤ 38 • at the 99% confidence level [6]). We can therefore conclude that a neutrino mass model predicting V 0 at Λ NP is in general disfavored in the MSSM.
This situation will change, if the generalized tri-bimaximal neutrino mixing pattern V ′ 0 is concerned. The reason is simply that the increase of θ 12 during its RGE running can be controlled by the Majorana phase factor cos 2 (α 2 − α 1 ), as shown in Eq. (7). Hence the proper fine-tuning of (α 2 − α 1 ) will allow θ 12 to mildly evolve into its experimental range 30 • ≤ θ 12 ≤ 38 • at Λ EW . Given tan β = 10, an approximate bound on (α 2 − α 1 ) is found to be 154 • ≤ |α 2 − α 1 | ≤ 206 • in our calculation. We present an explicit numerical example in Table II and Fig. 2, just for the purpose of illustration. As a direct consequence of θ 13 = 0 • at Λ NP , a very significant RGE correction to the CP-violating phase δ arises from the term proportional to 1/s 13 in dδ/dt. Thus δ can keep on staying at its quasi-fixed point in the MSSM, just like the case in the SM. Following the discussions given above, one may approximately arrive at the relations given in Eqs. (12) and (13) at the quasi-fixed point. The latter possibility has been ruled out by taking into account the evolution of θ 12 , and the former possibility is illustrated in Table II or Fig. 2, where the rather mild running behaviors of α 1 and α 2 can also be seen. In this example, θ 13 ≈ 1.4 • together with δ ≈ −78 • can be radiatively generated at Λ EW . This result implies 4 We have m ee = m 1 c 2 12 c 2 13 e iα 1 + m 2 s 2 12 c 2 13 e iα 2 + m 3 s 2 13 e −2iδ by using the parametrization of V given in Eq. (1). This result implies that it is actually ill to refer to δ as the Dirac phase in this "standard" parametrization. A different phase convention of V has been proposed in Ref. [24] to forbid δ to appear in m ee . that the magnitude of the Jarlskog invariant (i.e., J = s 12 c 12 s 23 c 23 s 13 c 2 13 sin δ) can be as large as about 0.6%, probably leading to the observable CP-violating effect in the future long-baseline neutrino oscillation experiments.
Let us stress that the quasi-fixed point in the RGE evolution of δ is in general unavoidable for those neutrino mixing patterns with θ 13 = 0. The above analytic understanding of such a quasi-fixed point is new and helpful for specific model building. If the phase convention of V in Eq. (1) is replaced by that proposed in [24], its corresponding Majorana phases ρ = δ + α 1 /2 and σ = δ + α 2 /2 will also have the quasi-fixed points in their RGE evolution. This point can easily be understood with the help of Eq. (8): the dominant term of dδ/dt (proportional to 1/s 13 ) will enter dρ/dt and dσ/dt, such that the RGE running behaviors of δ, ρ and σ are essentially identical [19]. In short, the "standard" parametrization of V taken in Eq. (1) is more convenient in discussing the issue of quasi-fixed points, while the phase convention of V advocated in Ref. [24] is more convenient in discussing the neutrinoless double-beta decay (i.e., m ee is dependent on ρ and σ but independent of δ). 4 We have argued that the tri-bimaximal neutrino mixing pattern V 0 or its generalized counterpart V ′ 0 is very likely to result from an underlying flavor symmetry, and this new symmetry is most likely to be realized at a superhigh energy scale. Supposing that this new physics scale is close to the neutrino seesaw scale (∼ 10 14 GeV), we have calculated the one-loop RGE effects on V 0 and V ′ 0 in their evolution down to the electroweak scale (∼ 10 2 GeV) in the working assumption that three neutrino masses are nearly degenerate. It is found that three mixing angles of V 0 or V ′ 0 are essentially insensitive to radiative corrections in the SM. In the MSSM, however, V 0 is in general disfavored and V ′ 0 can be compatible with current neutrino oscillation data if its two Majorana phases α 1 and α 2 are properly fine-tuned. We have also shown that it is possible to radiatively generate the CP-violating phase δ from α 1 and α 2 , and δ may keep on staying at its quasi-fixed point in both the SM and the MSSM.
Although a detailed RGE analysis of the tri-bimaximal neutrino mixing scenario has not been done before, some of our results have actually been observed by some other authors in their analyses of radiative corrections to the neutrino mass spectrum and realistic lepton flavor mixing patterns, whose forms are more or less similar to the tri-bimaximal neutrino mixing pattern V 0 . However, our present work is new in two important aspects: (1) we generalize V 0 to V ′ 0 with two arbitrary Majorana phases, because the latter is more interesting and can naturally appear in some specific neutrino mass models; (2) we explore the quasifixed point in the RGE evolution of δ and present an analytic understanding of this nontrivial phenomenon.
The afore-mentioned RGE running behaviors of V 0 and V ′ 0 are expected to be useful for model building at a superhigh energy scale. A similar study can be extended to some other interesting neutrino mixing patterns. For example, the pattern which has θ 12 = 30 • , θ 23 = 45 • and θ 13 = 0 • [25], is rather analogous to the generalized tribimaximal neutrino mixing pattern V ′ 0 . If U 0 is derived from an underlying flavor symmetry at an energy scale close to the seesaw scale, then its sensitivity to radiative corrections must be very similar to that of V ′ 0 . To examine whether such a special lepton mixing scenario is viable or not in a highenergy neutrino mass model, it is crucial to measure the smallest mixing angle θ 13 and the CP-violating phase δ in the future neutrino oscillation experiments. Furthermore, any experimental information about the Majorana phases α 1 and α 2 is welcome and extremely important, in order to distinguish one model from another through their different sensitivities to radiative corrections. TABLES TABLE I. Radiative corrections to V 0 (Case I) and V ′ 0 (Cases II and III) from Λ NP ∼ 10 14 GeV to Λ EW ∼ 10 2 GeV in the SM. The Higgs mass m H = 140 GeV has typically been input in our numerical calculation. Note that δ is not well-defined in the θ 13 = 0 limit at Λ NP , but its running behavior is independent of this ambiguity and is fixed by the initial values of α 1 and α 2 .

Parameter
Case  TABLE II. Radiative corrections to V ′ 0 from Λ NP ∼ 10 14 GeV to Λ EW ∼ 10 2 GeV in the MSSM. In our numerical calculation, tan β = 10 has typically been input. Note that δ is not well-defined in the θ 13 = 0 limit at Λ NP , but its running behavior is independent of this ambiguity and is fixed by the initial values of α 1 and α 2 .

Parameter
Input    Table II.