PRAMANA ­c Indian Academy of Sciences Vol. 54, No. 2 — journal of February 2000 physics pp. 235–246 Renormalization of seesaw neutrino masses in the standard model with two-Higgs doublets N NIMAI SINGH £ and S BIRAMANI SINGH Physics Department, Gauhati University, Guwahati 781 014, India £ Present address: Department of Physics and Astronomy, University of Southampton, Highfield, Southampton, S017 1BJ, UK Email: [email protected]; [email protected] MS received 28 August 1998; revised 12 August 1999 Abstract. Using the theoretical ambiguities inherent in the seesaw mechanism, we derive the new analytic expressions for both quadratic and linear seesaw formulae for neutrino masses at low en- ergies, with either up-type quark masses or charged lepton masses. This is possible through full radiative corrections arising out of the renormalizations of the Yukawa couplings, the coefficients of the neutrino-mass-operator in the standard model with two-Higgs doublets, and also the QCD–QED rescaling factors below the top-quark mass scale, at one-loop level. We also investigate numerically 8 b Å =¼:59 ¢ ½¼ the unification of top- - Yukawa couplings at the scale Á GeVforafixedvalue ¬ =58:77 of ØaÒ , and then evaluate the seesaw neutrino masses which are too large in magnitude to be compatible with the presently available solar and atmospheric neutrino oscillation data. How- ½½ Å =¼:59 ¢ ½¼ ever, if we consider a higher but arbitrary value of Á GeV, the predictions from linear seesaw formulae with charged lepton masses, can accommodate simultaneousely both solar atmospheric neutrino oscillation data. Keywords. Neutrino masses; radiative corrections; seesaw formula. PACS Nos 12.10; 13.15 1. Introduction The implication of any observational evidence on the possible non-zero neutrino mass, and neutrino oscillations, would indicate a departure from the standard model. Recent exciting results from the super-Kamiokande experiments [1] have reconfirmed the initial ¾ ³ data of atmospheric muon deficits, thereby suggesting a large mixing angle with ÆÑ Ü ¾ ¿ ½¼ (0.5–6)¢ eV . The SOUDAN-2 results [2,3] also support these findings with their ¾ ¾ ¿ ½ ¢ ½¼ value of ÆÑ being above eV . Further positive evidences pour from MACRO data [4] on upward-going muons. Using the recent CHOOZ results [5] which exclude ! e possibility of the oscillation , the super-Kamiokande result now just implies ! × the oscillation or a sterile neutrino (which is a singlet under the standard model). The collaboration using the liquid scintillator neutrino detector at Los Alamos 235 N Nimai Singh and S Biramani Singh ! ! e e (LSND) has reported evidences for the appearance of [6] and [6] oscil- ¾ ¾ ¾ :¾ ÆÑ ½¼ lations. Interpretation of the LSND data favours the choice ¼ eV eV for ¾ :¼¼¾ ×iÒ ¾ ¼:¼¿ ¼ . However the KARMEN-2 experiment [7] which is also sensitive to this region of parameter space, restricts the allowed values to a relatively small subset of the above region. There are also indications of neutrino oscillations in the neutrino com- ing from the sun. The solar neutrino puzzle can be resolved [8] through matter enhance ¾ ¾ ¾ 5 ³ ¢½¼ ×iÒ ¾ ½ oscillation (MSW) with ÆÑ (0.8–2) eV , , or vacuum oscillation eÜ ¾ ¾ ¾ ½¼ ³ ¢½¼ ×iÒ ¾ ½ ÆÑ (0.5–6) eV , . The Sudbury Neutrino Observatory (SNO) [9] eÜ and BOREXINO [10] under construction, are expected to play major role in future. There are also other indications of the non-zero mass of the neutrinos. Neutrino mass in the range of (1–6) eV may provide viable candidate for the hot dark matter (HDM) component of the Ñ < ¼:46 universe [11,12], and Majorana neutrino mass e eV may give positive evidence for the neutrinoless double beta decay [13–15]. Theoretically, Gell-Mann, Ramond and Slansky [16], and Yanagida [17] were the first authors to point out that small left-handed Majorana neutrino masses can naturally arise through seesaw mechanism in which a large mass of the right-handed Majorana neutrino is associated with the spontaneous symmetry breaking of the left-right symmetric grand unified theories such as SO(10) GUT where the gauge couplings of the standard model are unified (i.e. grand desert model). However, the grand desert models are ruled out by experimental constraints on electroweak mixing angle and proton lifetime. But predictions from SO(10) GUT are consistent with experimental data provided intermediate symmetries ¢ ¢ ´= G Ê C ¾¾4 such as SU(2) Ä SU(2) SU(4) ), exist in the model as in case of two-step breaking of SO(10) [18–20]. In order to determine the allowed value of the intermediate Å ´4µ C symmetry breaking scale Á at which SU is broken in a class of SO(10) models, we usually have two independent methods. One method consists of running the gauge Å couplings and using the matching conditions at Á . The other is by demanding that the Å Yukawa couplings of the third generation get unified at Á . Since SO(10) contains the G maximal subgroup ¾¾4 , the simplest and most attractive assumption about the scale at C which SU(2) Ê breaks is that it is the same scale at which SU(4) breaks. This would be the case if the intermediate symmetry breaking were done by the right-handed neutrino Å Å Á masses Æ . Such a two-step SO(10) model is particularly promising in the sense that it includes the seesaw mechanism in a natural way and predict small neutrino masses. The above consideration is generally valid to other groups than SO(10) where SU(4) C and SU(2) Ê break at the same scale [18]. In this paper we follow the second option in which the top-b- Yukawa couplings unify Å at Á , and this will be a consequence of the low-energy data and two-Higgs doublet stan- dard model (SM) which may emerge from SO(10) GUT. Freire [18] had investigated the unification of three Yukawa couplings of the third generation in SO(10) with two-Higgs- ½¾:¾ ½¿:6 Å ´½¼ ½¼ µ doublets SM and observed the intermediate scale at Á GeV for ØaÒ ¬ 8¼ Ñ ½8¼ 35–45, yielding the bounds, GeV ØÓÔ GeV. Subsequently Parida 8:5 9:5 Å ´½¼ ½¼ µ and Usmani [20] investigated quark-lepton unification scale at Á – GeV Ñ = ØaÒ ¬ for ØÓÔ (160–190) GeV with 52.80–261.94, respectively. In the light of the precise value of top-quark mass from TEVATRON, the question of quark-lepton uni- Å fication with the accurate value of Á , and generation of small neutrino masses through seesaw mechanism valid at the intermediate scale, can be reexamined, and compared with the recent observations on solar and atmospheric neutrino oscillations. We organize the paper in the following way. Section 2 is devoted to the derivation of the 236 Pramana – J. Phys., Vol. 54, No. 2, February 2000 Seesaw neutrino masses low-energy radiative corrections to both quadratic and linear seesaw formulae for Majorana neutrino masses in the standard model with two-Higgs doublets, which might emerge from SO(10) GUT. In Ü3 we outline the procedure for the numerical analysis and present the main results. The last section is devoted to summary and conclusion. 2. Derivation of analytic expressions for neutrino masses The theoretical ambiguities occurring in the seesaw mechanism lead to four types of linear Ñ and quadratic seesaw models [21] with Dirac-type fermion masses i are either up-quark Ñ ;Ñ ;Ñ Ñ ;Ñ ;Ñ c Ù e masses ( ØÓÔ ) or charged lepton masses ( ). In case of the familiar quadratic seesaw formula, the right-handed neutrino masses are assumed to be degenerate ¾ Å Ñ Ñ = Å Æ Æ , and the left-handed Majorana neutrino masses vary as (and hence i i i quadratic), ¾ Ñ ´Å µ Á i Ñ ´Å µ= ; Å Å ; i=½;¾;¿: Æ Á Á (1) i Å Æ Å Ñ i Alternatively, the eigenvalues of Æ may follow the same hierarchy as in case of non- i Å Ñ i degenerate right-handed neutrino mass (i.e. Æ is approximately proportional to ). i Ñ Ñ i This leads to linear seesaw formula where vary as [21], i Ñ ´Å µ i Á : Ñ ´Å µ=Ñ ´Å µ Á i Á (2) i Å Æ i Ñ = h Î Î = Î ×iÒ ¬ Î = Î cÓ× ¬ i Ù;d Ù d Here i ,where for up-type quarks and for charged Å = f Î f ³ h Å i Ê i i Á leptons; and Æ .If at the intermediate scale , then the linear seesaw i formula (2) becomes Î ´Å µ Ù;d Á Ñ ´Å µ ³ Ñ ´Å µ ; Á i Á (3) i Î Ê = ½74 where the vacuum expectation values are taken as Î GeV at the top-quark mass Î Å = Å Æ Á scale, and Ê . The left-handed Majorana neutrino mass matrix can have ½ ¾ Ñ = à Πà i =½;¾;¿ ii ii the form [22], where ( ) is the coefficient of the dimension i 4 ½ ½ Ä Ã ii five neutrino-mass-operator, , and has a dimension of [mass] .The 4 neutrino mass ratio due to the renormalization effect, can be estimated as ¾ Ñ ´Ø µ à ´Ø µ Î ´Ø µ Á ii Á Á i = ¢ ; Ñ ´Ø µ à ´Ø µ Î ´Ø µ ¼ ii ¼ ¼ i Ø = ÐÒ´Ñ =½ µ Ø = ÐÒ´Å =½ µ ØÓÔ Á Á where ¼ GeV and GeV . Similarly, the Dirac-type fermion mass ratios are also estimated as Ñ ´Ø µ h ´Ø µ Î ´Ø µ i Á i Á Ù;d Á = ¢ : Ñ ´Ø µ h ´Ø µ Î ´Ø µ i ¼ i ¼ Ù;d ¼ Ñ Ñ i With the above radiative corrections to and , the quadratic and linear seesaw for- i Ø mulae in eqs (1), (3) can be expressed at the top-quark mass scale ( ¼ ) Pramana – J.