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Numerical consistency check between two approaches to radiative corrections for neutrino masses and mixings

We briefly outline the two popular approaches on radiative corrections to neutrino masses and mixing angles, and then carry out a detailed numerical analysis for a consistency check between them in MSSM. We find that the two approaches are nearly consistent with a small discrepancy of a factor of 13 percent in mass eigenvalues at low energy scale, but the predictions on mixing angles are almost consistent. We check the stability of the three types of neutrino models, i.e., hierarchical, inverted hierarchical and degenerate models, under radiative corrections, using both approaches, and find consistent conclusions. The neutrino mass models which are found to be stable under radiative corrections in MSSM are the normal hierarchical model and the inverted hierarchical model with opposite CP parity. We also carry out numerical analysis on some important conjectures related to radiative corrections in MSSM, viz., radiative magnification of solar and atmospheric mixings in case of nearly degenerate model having same CP parity (MPR conjecture) and radiative generation of solar mass scale in exactly two-fold degenerate model with opposite CP parity and non-zero reactor angle (JM conjecture). We observe certain exceptions to these conjectures. Finally the effect of scale-dependent vacuum expectation value in neutrino mass renormalisation is discussed.


Introduction
Recent developments in the determination of neutrino masses and mixing angles from various oscillation experiments, have strengthened our knowledge of neutrino physics [1]. In order to have a meaningful comparison of the theoretical predictions on neutrino masses and mixing angles within the framework of GUTs with or without supersymmetry, with the datas from various neutrino oscillation experiments [2], the effects of radiative corrections are very essential [3]. Considerable progress has been achieved at this front and this can be mainly classified into two categories -(i) the evolution of the renormalisation group equations (RGEs) from high to low energy scale [4,5,6,7], and (ii) the low energy threshold corrections [8]. In case of running the RGEs in (i), the general underlying motivations are : to check the stability of the neutrino mass model under radiative corrections [9,10]; to generate solar mass scale and also reactor angle |U e3 | from radiative corrections [11]; to generate correct radiative magnifications of solar and atmospheric mixing angles from CKM-like small values at high scale [12]; and to get suitable deviations from the bimaximal solar and atmospheric mixings through radiative corrections [13,14], etc. One basic difference of the first one from the last three points is the very definition of the stability criteria under radiative corrections [15]. Radiative stability generally means that the effects of radiative corrections do not alter substantially the good predictions on neutrino masses and mixings already acquired through seesaw mechanism at high scale M R .
Within the framework of running the RGEs from high energy scale to low energy scale, we have again two different approaches so far employed in the literature. In the first approach (we call 'Method A' for simplicity) the running is carried out through the neutrino mass matrix m LL as a whole, and at every energy scale one can extract neutrino masses and mixing angles through the diagonalisation of the neutrino mass matrix calculated at that particular energy scale [6,7,10]. In the second approach (we call 'Method B') the running of the RGEs is carried out in terms neutrino mass eigenvalues and three mixing angles [16,17,18]. We confine our analysis to CP-conserving case, neglecting all CP phases in the neutrino mixing matrix.
In the present paper we carry out a detailed numerical analysis of these two aprroaches for a consistency check on numerical accuracy, and find out the stability criteria of the main three neutrino mass models [19]. We give all the zeroth-order as well as full textures of the left-handed neutrino mass matrices obtained from seesaw mechanism in Appendix (A,B), and use these ex-pressions for checking the stability criteria. In addition, we further study the validity of some existing conjectures based on radiative corrections. The effect of running vacuum expectation value to the evolution of neutrino masses, is further examined in both approaches. The paper is organised as follows.
In section 2, we briefly outline the main points of the two approaches on renornalisation group analysis. The numerical analysis and main results are presented in section 3. Section 4 concludes with a summary and discussion.

Renormalisation group analysis of neutrino masses and mixings
We present a very brief review on the two main approaches of taking quantum radiative corrections of neutrino masses and mixings in MSSM. Our main motivation is to have a numerical consistency check on the results of these two approaches, and apply again to check the validity on some existing conjectures related to radiative corrections.

Method A: Evolution of neutrino mass matrix
In this approach the quantum radiative corrections are taken on all the elements of the neutrino mass matrix m LL in the basis where charged lepton mass matrix is diagonal. The diagonalisation of the neutrino mass matrix at any particular energy scale leads to the physical neutrino mass eigenvalues as well as three mixing angles. The neutrino mass matrix m LL (t) which is generally obtained from seesaw mechanism, is expressible in term of K(t), the coefficient of the dimension five neutrino mass operator in a scale-dependent manner, t = ln(µ/1GeV ), where the vacuum expectation value (vev) is v u = v 0 sin β, and v 0 = 174GeV in MSSM. The evolution equation of the coefficient K(t) in the basis where the charged lepton mass matrix is diagonal, is given by [10] d dt We replace K(t) by m LL (t) in the above equation where we assume that vev is scale-independent. Upon integration from high (B − L) breaking scale Here the overall factor R 0 which does not affect the mixing angles, is given by the expression, which strongly depends on top-quark Yukawa coupling and gives very large contribution for small tan β value (=1.60). The integrals in the above expressions are defined as [7,10] I gi (t 0 ) = 1 16π 2 and, where i = 1, 2, 3 and f = t, b, τ respectively. The numerical values of these integrals at different energy scales, can be calculated from the running of the RGEs for three gauge couplings (g 1 , g 2 , g 3 ) and the third family Yukawa couplings (h t , h b , h τ ) in MSSM. The mass eigenvalues and the MNS mixing matrix [20] are estimated through diagonalisation of m LL at every point in the energy scale, and where f = τ, µ, e and i = 1, 2, 3. The MNS mixing matrix [20] is usually parametrised in term of three rotations (neglecting CP violating phases) by where s ij = sin θ ij and c ij = cos θ ij respectively. The unitarity conditions are also satisfied Hence tan θ 12 = |U e2 |/|U e1 | and tan θ 23 = |U µ3 |/|U τ 3 | and sin θ 13 = |U e3 |. The solar LMA MSW solution favours the 'light-side' tan θ 12 < 1 of the data [21,22,23] for the usual sign convention |m 2 | > |m 1 |. It is also possible to express mixing angles in terms of sin θ ij directly [24].

Method B: Evolution of neutrino mass eigenvalues and mixing angles
Here we follow the main expressions from Ref. [16] for the evolution of three neutrino mass eigenvalues and mixing angles 2 . In this approach the RGEs for the eigenvalues of coefficient K(t) can be expressible as Neglecting h 2 µ and h 2 e compared to h 2 τ , and taking the scale-independent vev as before, we have the complete RGEs for three neutrino mass eigenvalues [16], One can have an approximate analytical solution of the above equation by neglecting the small effect due to the change of U 2 τ i in the integration range, as [15] These equations lead to the equations derived in JM conjecture [11] and have interesting consequences. However it should be emphasised that these equations are valid when the mixing angles are only static. The corresponding evolution equations for the MNS matrix elements U f i are given by [16] where f = e, µ, τ and i = 1, 2, 3 respectively. Neglecting h 2 µ and h 2 e as before, one can simplify the terms, can write the RGEs for all the elements of MNS matrix. For example, we give here only three of them relevant to our requirement, Using the MNS parametrisation in eq.(10), the above three expressions(15) -(17) simplify to [16]

Effect of scale-dependent vev
In this section we modify the results of Ref. [16] by considering the running of the vev v u (t) through the neutrino mass formula m i (t) = v 2 u (t)K i (t). This gives [15] d where the RGE for v u in MSSM is given by The complete RGEs for neutrino mass eigenvalues are now given by [15] d dt leading to the approximate solution, Similarly, for Method A with the inclusion of scale-dependent vev, R 0 in eq.(4) is now replaced by Now the top-quark dependent term has been cancelled out and this will certainly affect the overall magnitude of the neutrino masses, but not the mixing angles. With this modification the magnitudes of neutrino masses tend to increase with the decrease of energy scale. The enhancement factor in the magnitude of neutrino masses through RGEs is calculated as which gives a positive numerical value greater than one even for large tan β values (= 55).

Numerical analysis and results
For a complete numerical analysis of the RGEs for both Methods A and B presented in the previous section, we follow here two steps (a) bottom-up running [7] and (b) top-down running [10]. In the first step (a), the running of the RGEs for the third family Yukawa couplings (h t , h b , h τ ) and three gauge couplings (g 1 , g 2 , g 3 ) in MSSM , is carried out from top-quark mass scale (t 0 ) at low energy end to high energy scale M R where B − L symmetry breaks down [7,15]. In the present analysis we consider the high scale M R = 10 13 GeV and take the large tan β input value ( tan β = 55). For simplicity of the calculation, we assume here the supersymmetric breaking scale at the top-quark mass scale t 0 = ln m t [7]. We adopt the standard procedure to get the values of gauge couplings at top-quark mass scale from the experimental CERN-LEP measurements at M Z , using one-loop RGEs, assuming the existence of a one-light Higgs doublet and five quark flavours below m t scale [15]. Similarly, the Yukawa couplings are also evaluated at top-quark mass scale using QCD-QED rescaling factors in the standard fashion [15]. We present here the values of the Yukawa couplings and gauge couplings at two scales t 0 = 5.159 and t R = 29.954 as follows: The values of the integrals are estimated between the two limits (t 0 , t R ). In the second stage (b), the running of three neutrino masses (m 1 , m 2 , m 3 ) and mixing angles (s 12 , s 13 , s 23 ) is carried out together with the running of the Yukawa and gauge cuoplings, from the high scale t R to low scale t 0 . In this case we use the values of Yukawa and gauge couplings evaluated earlier at the scale t R from the first stage runnung of RGEs. In principle one can evaluate neutrino masses and mixing angles at every point in the energy scale.
We present the results of our numerical analysis in Tables 1-4 are slightly reduced, whereas the mixing angles are slightly increased with the decrease in energy scale. But there is no substantial change in both parameters, which may spoil the good predictions already achieved at the high energy scale. In case of inverted hierarchical model with same CP parity ( IIA), the solar angle (s 12 ) is not stable under radiative corrections. Similarly, in all the three degenerate models (I-A,B,C) both solar and atmospheric mass scales as well as solar and atmospheric angles, are not stable under radiative corrections.
In Table 2 we analyse the MPR conjecture [12] which states that quark and lepton mixing angles are identical at high energy scale, and large solar and atmospheric neutrino mixing angles together with the small reactor angle, can be understood purely as a result of RG evolution provided the three neutrino masses are quasi-degenerate and have the same CP parity. In the present numerical analysis in Method B, both Yukawa and gauge couplings are running together with neutrino masses and mixings. It appears that in MPR paper [12] only neutrino masses and mixings are running. We present here three set of readings ( cases (i),(ii),(iii)) which give good radiative magnifications. However in case (iv) we use the same input values quoted in MPR paper but we observe less magnification contrary to MPR paper [12]. This discrepancy may be due to the running of gauge and Yukawa couplings along with neutrino masses and mixings in the form of coupled equations in the present analysis. In short, MPR conjecture [12] is well verified. The analysis presented here in Table 2 is the results of Method B, but we find that both methods (A and B) give consistent results.
In Table 3 we supply some new results connected to two more conjectures on radiative corrections in MSSM. JM conjecture [11] specifies that radiative corrections can generate the neutrino mass-squared difference required for the large mixing angle (LMA) MSW solution to the solar neutrino problem if two of the three neutrino masses are assumed to be exactly degenerate (m, −m, m ′ ) at high energy scale, and also if U e3 at high scale is non-zero. We have shown in Table 3 that for a limited range of non-zero values of m ′ , it is also possible to generate LMA MSW solution even if U e3 is zero at high scale. If both m ′ and U e3 are zeros at high scale, then it is not possible to generate LMA MSW solution. Case (ii) in Table 3 is interesting in the sense that it gives exception to JM conjecture by taking U e3 = 0 and m ′ = 0.07eV . At low energy we get △m 2 12 = 5.23 × 10 −5 eV 2 and sin θ 12 = 0.7042421 which is high but lies in the light side of the data (tan θ 12 < 1) [21,22,23]. Fig.1 shows the evolution of the three mixing angles, and the CHOOZ angle sin θ 13 is generated through radiative corrections. Fig.2 presents the evolution of the three neutrino masses, and the solar mass scale generated through radiative corrections at low scale, is then demonstrated in Fig.3. It is still necessary to tone down the solar mixing angle through further fine tuning. As an example, using s 12 = 0.7 at high scale, one can get s 12 = 0.697 at lower scale. We also discuss MST conjecture [14] which states that starting from bimaximal mixings at high scale, radiative corrections due to the τ -Yukawa coupling leads to solar angle towards the dark side at low scale,(tan θ 12 > 1). The results presented in Table 3 also show that MST conjecture [14] is not always valid for the cases (i)-(iv) discussed here.
Finally, as expected, the effect of running vev in neutrino mass formula leads to the increase of neutrino masses with the decrease of energy scale [15]. In Table 4 we calculate the neutrino masses and mixings with (I) or without (II) the effect of running the vev, at low energy scale. There is a factor of about 1.91 higher in case of low-energy neutrino mass values obtained with the running effect of vev. Figs. 4 and 5 show the evolution of neutrino masses for the above two cases (II) and (I) respectively.   Table  2: Analysis on MPR conjecture [12] related to radiative magnification on solar and atmospheric mixing angles at low scale. The parameters (m 0 1,2,3 , s 0 23,13,12 ) are defined at high scale M R = 10 13 GeV and others are defined at low scale m t = 175GeV. Cases (i)-(iv) include different sets of arbitrary input parameters.  Table 3: Analysis on JM conjecture [11] related to radiative magnification on solar mass scale at low scale. The parameters (m 0 1,2,3 , s 0 23, 13,12 ) are defined at high scale M R = 10 13 GeV and others are defined at low scale m t = 175GeV. Cases (i)-(iv) include different sets of arbitrary input parameters.

Summary and Discussion
We summarise the main points in this work. First we briefly review the main points of the formalism based on two approaches on the evolution of RGEs of neutrino masses and mixings. The first one (A) deals with the running of the whole neutrino mass matrix from which one can extract mass eigenvalues and mixings at any particular energy scale, whereas in the second approach (B) the three neutrino mass eigenvalues and the three mixing angles are running directly. Detailed numerical analysis shows that both approaches agree up to a discrepancy of a factor of 13% in mass eigenvalues. The predictions on mixing angles are almost consistent in these two approaches. Using both approaches we show that hierarchical model(III) and inverted hierarchical model with opposite CP parity (IIB) are stable under radiative corrections. The evolution of sin θ 12 is very fast in the inverted hierarchical model with same CP parity (IIA), and hence the model is not stable. We also verify the MPR conjecture [12] in which radiative magnification of solar and atmospheric mixings are possible in case of nearly degenerate model with same CP parity. We find that runnings of masses and mixings with and without the running of gauge and Yukawa couplings, give reasonably different magnifications. However such radiative amplification generally involves a delicate fine-tuning of the initial conditions which are to some extent unnatural [17]. However such problems are not there in the theory of neutrino masses derived from Kahler potential in supersymmetric model, and neutrino mixings angles can easily be driven to large values at low energy as they approach infrared pseudo-fixed points at large mixing [17].
We also study JM conjecture [11] which specifies the radiative generation of solar scale in exactly two-fold degenerate model having opposite CP parity (m, −m, m 3 ) and non-zero values of U e3 . Our numerical analysis shows that the same radiative generation of solar mass scale is also possible with the conditions U e3 = 0.0 and non-zero value of m 3 . We also discuss the MST conjecture [14] which states that starting from bimaximal mixings at high scale, radiative corrections lead to the solar angle towards the dark side of the data at low energy scale. We show that MST conjecture is not always valid. We suggest further generalisation of JM conjecture and this will be reported in subsequent communication [25]. Finally, the effect of running the vev in neutrino mass renormalisation is discussed and it is observed that neutrino mass increases with the decrease of energy scale when we include the running of vev. This gives magnification of a factor of 1.91 in neutrino masses at low energy scale, compared to the values calculated without the running vev effect. Numerical analysis in three generations with arbitrary CP violating phases is of great interest [18] and there has been a possibility that within a restricted range of the physical parameters including phases, the degenerate models are found to be stable under radiative corrections in MSSM [26]. As emphasised before, oversimplifications of analytic expressions and a departure from the simultaneouse running of Yukawa and gauge couplings along with neutrino masses and mixings, may have a danger of getting misleading conclusions.