Renormalization of Seesaw Neutrino Masses in the Standard Model with Two-Higgs Doublets

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 energies, 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

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

¿ ½¼

(0.5–6)¢ eV . The SOUDAN-2 results [2,3] also support these ﬁndings with their

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 ÆÑ ½¼

lations. Interpretation of the LSND data favours the choice ¼ eV eV for

: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 standard 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

Å Ñ Ñ = Å

Æ , and the left-handed Majorana neutrino masses vary as (and hence

i i i

quadratic),

Ñ ´Å µ

Ñ ´Å µ= ; Å Å ; i=½;¾;¿:

Æ Á

Á (1)

Å Ñ i

Alternatively, the eigenvalues of Æ may follow the same hierarchy as in case of non-

Å Ñ i

degenerate right-handed neutrino mass (i.e. Æ is approximately proportional to ).

Ñ Ñ i

This leads to linear seesaw formula where vary as [21],

Ñ ´Å µ

i Á

: Ñ ´Å µ=Ñ ´Å µ

Á i Á

(2)

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)

= 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

Ñ = Ã Î Ã i =½;¾;¿

ii ii

the form [22], where ( ) is the coefﬁcient of the dimension

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

Ñ ´Ø µ Ã ´Ø µ Î ´Ø µ

Á ii Á Á

= ¢ ;

Ñ ´Ø µ Ã ´Ø µ Î ´Ø µ

¼ ii ¼ ¼

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. Phys., Vol. 54, No. 2, February 2000 237

N Nimai Singh and S Biramani Singh

Ñ ´Ø µ

Ñ ´Ø µ=C ; ´Ø µ

¼ (4)

i i

Ã ´Ø µ h ´Ø µ

ii ¼ i Á

¢ C ´Ø µ=

h ´Ø µ Ã ´Ø µ

i ¼ ii Á

for the quadratic seesaw formulae, and

Î ´Ø µ

Ù;d ¼

Ñ ; ´Ø µ=C ´Ø µÑ ´Ø µ

¼ i ¼

¼ (5)

i i

Ã ´Ø µ h ´Ø µ

ii ¼ i Á

¢ C ´Ø µ=

h ´Ø µ Ã ´Ø µ

i ¼ ii Á for the linear seesaw formulae, respectively. In eqs (4), (5), the effect of radiative correc-

tions arising from the renormalization of the vacuum expectation value (Î ) cancels on both

sides of the equations. C

Since the radiative correction coefficients involve the ratios of the Yukawa cou- i plings, and also the coefficient of neutrino-mass-operator, we first collect all the relevant

renormalization group equations (RGEs) for these quantities in two-Higgs doublet stan-

Ñ Å Á dard model, in the energy range ØÓÔ . One-loop RGEs for Yukawa couplings

of quarks and leptons are given by [23–25]

" #

9 dh ½

ØÓÔ

¾ ¾ ¾ ¾

½6 = h h h · C g ;

ØÓÔ i

ØÓÔ b i

dØ ¾ ¾

" #

9 ½ dh

¾ ¾ ¾ ¾ ¼ ¾

= h h h · · h ½6 C g ;

b ØÓÔ i i

dØ ¾ ¾

" #

5 dh

¾ ¾ ¼¼ ¾ ¾

= h h ·¿h C ½6 g ;

b i i

dØ ¾

" #

Ù;c

¾ ¾ ¾

= h ¿h C g ; ½6

Ù;c i

ØÓÔ i

dØ

" #

¾ ¾ ¾ ¼¼ ¾

= h ¿h · h ½6 C g ;

;e (6)

b i i

dØ

Ø = ÐÒ´=½ µ h = j

where GeV , j Yukawa coupling of the th fermion, and

½ ½ ¼ ¼

½7=¾¼ 9=4 8 C

A A @ @

½=4 9=4 8 C

; i = Y; ¾Ä; ¿C: =

¼¼

9=4 9=4 ¼ C

´g =4 = « µ

The two-loop RGEs for the evolution of gauge couplings i are given by [23]

¿ ¾

¾ ¾ ¾ ¾ ¾ ¾

5 4

; · g ´½6 µ = g ½6 b g b g

i ij

i (7)

j i i

dØ

j =½ where

238 Pramana – J. Phys., Vol. 54, No. 2, February 2000

Seesaw neutrino masses

b = ´¾½=5; ¿; 7µ;

½ ¼

½¼4=¾5 ½8=5 ½44=5

A @

8=5 8 ½¾

: b =

ij (8)

½½=½¼ 9=¾ ¾6

In order to get a closed form analytic solution, the dominant part of RGE for the coefﬁcient

of the neutrino-mass-operator can be approximated as [22,26]

¢ £

dÃ

¿¿

¾ ¾ ¾ ¾

½6 ³ Ã 6h · h ·¾ ¿g ;

¿¿ ¾

ØÓÔ ¾

dØ

¢ £

dÃ

¾¾;½½

¾ ¾ ¾

½6 ³ Ã 6h ·¾ ¿g ;

;½½ ¾

¾¾ (9)

ØÓÔ ¾

dØ

= ½½; ¾¾; ¿¿

where (ii ) refer to the family indices, and the evolution equations for the

´ ; ; ; ; µ

¾ ¿ 4 5

ﬁve quartic-Higgs scalar coefﬁcients ½ are given in paper by Hill et al

Ø Ø Ø Á

[27]. Following the standard method of integration in the energy range ¼ where

Ø = ÐÒ´Ñ =½ µ Ø = ÐÒ´Å =½ µ Ø = ÐÒ´=½ µ

ØÓÔ Á Á

¼ GeV , GeV ,and GeV , eqs (6), (9) reduce to

[h ´Ø µ=h ´Ø µ] [Ã ´Ø µ=Ã ´Ø µ] Á Á

Á i ¼ ii ¼ ii Á i

i and which are expressible in terms of integrals , ,

[« ´Ø µ=« ´Ø µ]

Á i ¼

and ratios i where

h ´Øµ

Á = dØ; i = ; b; ØÓÔ;

½6

´Øµ

Á = dØ; i =½;¾;¿;4;5:

(10)

½6

= ´Ñ ´Ñ µµ=

i i

The QCD–QED rescaling factors for fermion masses, deﬁned by [23] i

´Ñ ´Ñ µµ ØÓÔ i , are estimated following the standard techniques at three-loop order in QCD. These factors take care of the renormalization of fermion masses from top-mass scale down to the respective low-energy scale of each Dirac-fermion mass in question. Collecting all pieces, the standard seesaw formulae (4), (5) can be presented in four different types.

Type I: Quadratic seesaw formula (QSF) with up-type quark masses

The low energy neutrino masses are given by

ØÓÔ

Ñ ´Ø µ=C ´Ø µ ;

¼ ¼

Ñ ´Ø µ=C ´Ø µ ;

c c

; Ñ ´¼µ = C ´¼µ

(11)

e e

Ø = ÐÒ´Ñ =½ µ Ø = ¼ = ÐÒ´½ =½ µ C

c ½

where c GeV , GeV GeV , and the coefﬁcients are i calculated as

Pramana – J. Phys., Vol. 54, No. 2, February 2000 239

N Nimai Singh and S Biramani Singh

C ´Ø µ=Ê ´Ø µ eÜÔ [¿Á · Á Á ¾Á ] ;

¼ Á ¼ ØÓÔ b

Ê ´Ø µ

Á ¼

eÜÔ [ ¾Á ] ; C ´Ø µ=

Ê ´Ø µ

Á ¼

eÜÔ [ ¾Á ] ; C ´¼µ =

(12)

¾ e

with

½i

« ´Ø µ

i Á

Ê ´Ø µ= ; Ô =´ ½7=84; ½=4; 8=7µ:

Á ¼ ½i

« ´Ø µ

i ¼

i=½

Type II: Quadratic seesaw formula (QSF) with charged lepton masses

In this case, the quark masses in formulae (12) are replaced by corresponding charged

leptons masses

Ñ ´Ø µ=C ´Ø µ ;

Ñ ´Ø µ=C ´Ø µ :

e e

(13)

e e

Æ C

The analogous expressions for deﬁned at the respective lepton mass scales, are worked i

out as

Ê ´Ø µ

ÁÁ ¼

C ´Ø µ= eÜÔ [Á ·6Á Á ¾Á ];

ØÓÔ b

Ê ´Ø µ

ÁÁ ¼

C ´Ø µ= eÜÔ [ 6Á ·6Á ·¾Á ¾Á ];

ØÓÔ b

Ê ´Ø µ

ÁÁ ¼

eÜÔ [ 6Á ·6Á ·¾Á ¾Á ]; C ´Ø µ=

ØÓÔ b e

(14)

¾ e

¾i

« ´Ø µ

i Á

Ê ´Ø µ= ; Ô =´ ½5=¾8; ½=4; ¼µ;

ÁÁ ¼ ¾i

« Ø µ

i ¼

i=½

Ø = ÐÒ´Ñ =½ µ i = ; ; e i where i GeV , .

Type III: Linear seesaw formula (LSF) with up-type quark masses

The neutrino masses at low energies deﬁned at the respective fermion mass scales are now given by

240 Pramana – J. Phys., Vol. 54, No. 2, February 2000

Seesaw neutrino masses

Ñ Î

ØÓÔ Ù

Ñ ; ´Ø µ=C ´Ø µ

¼ ¼

Ñ Î

c Ù

Ñ ´Ø µ=C ; ´Ø µ

c c

Ñ Î

Ù Ù

Ñ ; ´¼µ = C ´¼µ

(15)

e e

Ê C

with the expressions for at low energies,

½ ¿

C Á · Á Á ¾Á ´Ø µ=Ê ´Ø µ eÜÔ ;

ØÓÔ b ¼ ÁÁÁ ¼

¾ ¾

Ê ´Ø µ

ÁÁÁ ¼

C eÜÔ [ ¿Á ¾Á ´Ø µ= ] ;

ØÓÔ c

Ê ´Ø µ

ÁÁÁ ¼

eÜÔ [ ¿Á ¾Á C ] ; ´¼µ =

ØÓÔ

(16)

¾ e

where

¿i

« ´Ø µ

i Á

Ê ´Ø µ= ; Ô =´ ½7=½68; ½=8; 8=½4µ:

ÁÁÁ ¼ ¿i

« ´Ø µ

i ¼

i=½

Type IV: Linear seesaw formula (LSF) with charged lepton masses

Deﬁning neutrino masses at the respective mass scales of the charged leptons,

Ñ Î

Ñ ´Ø µ=C ´Ø µ ;

Ñ Î

Ñ ´Ø µ=C ´Ø µ ;

Ñ Î

e d

; Ñ ´Ø µ=C ´Ø µ

e e

(17)

e e

Ê C

the coefﬁcients are now expressed by

Ê ´Ø µ 7

ÁÎ ¼

C ´Ø µ= eÜÔ Á ·¿Á ·Á ¾Á ;

ØÓÔ b

Ê ´Ø µ

ÁÎ ¼

eÜÔ [ 6Á ·¿Á ·Á ¾Á ]; C ´Ø µ=

ØÓÔ b

Ê ´Ø µ

ÁÎ ¼

C ´Ø µ= eÜÔ [ 6Á ·¿Á ·Á ¾Á ];

e b

ØÓÔ (18)

e ¾

« ´Ø µ

i Á

Ê ´Ø µ= ; Ô =´ 45=½65; ½=8; ¼µ:

ÁÎ ¼ 4i

« ´Ø µ

i ¼

i=½ C

While deriving the expressions for in all four types (I–IV) of seesaw formulae, i family mixing based on certain texture is not considered for simplicity. The right-handed

Pramana – J. Phys., Vol. 54, No. 2, February 2000 241

N Nimai Singh and S Biramani Singh

Å ³ Î = Å

Ê Æ

Majorana neutrino mass scale is taken as Á . One can also express the

C Ñ =C =Ñ

ratios and in all four types of seesaw formulae in order to show the

;e ;e hierarchical structures of the three neutrino species.

3. Numerical solution and results

C Ñ

As necessary ingredients for the numerical estimation of and in eqs (11)–(18),

i i

g h i

our next step is to solve the RGEs in eqs (6), (7), (9) for i and at different points of

Ñ Å Á

Á i

energy scale, ØÓÔ , using the standard Runge-Kutta method. The integrals Å

deﬁned in eq. (10) are then evaluated for the ﬁxed Á which shall be determined by the

´h ;h ;h µ ØaÒ ¬

b uniﬁcations of Yukawa couplings, ØÓÔ for a certain ﬁxed value of . The input values of the Yukawa couplings at the top-quark mass scale, are given by

[28,29]

h ´Ñ µ=Ñ ´Ñ µ=´½74 ×iÒ ¬ µ;

ØÓÔ ØÓÔ ØÓÔ ØÓÔ

h ´Ñ µ=Ñ ´Ñ µ=´½74 cÓ× ¬ µ;

b ØÓÔ b b b

h ´Ñ µ=Ñ ´Ñ µ=´½74 cÓ× ¬ µ;

ØÓÔ

(19)

Ñ ´Ñ µ Ñ ´Ñ µ Ñ ´Ñ µ

ØÓÔ b b where ØÓÔ , ,and are the running masses. For heavy ﬂavours of quark, the difference between pole (expt.) and running masses are quite signiﬁcant. The

most recent determination of the input parameters are [30,31]

Ñ ´Ñ µ = ½66:5 ; Ñ = ½:785 ;

ØÓÔ

ØÓÔ GeV GeV

Ñ ´Ñ µ = 4:¾ ; Ñ = ¼:½¼5 ;

b b

GeV GeV

Ñ = ½:¾5 ; Ñ = ¼:¼¼¼5 ; e

c GeV GeV

Ñ = ¼:¼¼5 ;

Ù GeV

where the running masses of top and bottom quarks are obtained from the respective pole

Ô ÓÐe Ô ÓÐe

= ½75:6 Ñ = 4:7

masses Ñ GeV and GeV, respectively, at two-loop level. The

ØÓÔ b

CERN-LEP measurements are given by

×iÒ ´Å µ=¼:¾¿½6 ¦ ¼:¼¼¼¿;

! Z

« ´Å µ=¼:½½8 ¦ ¼:¼¼4;

× Z

« ´Å µ = ½¾7:9 ¦ ¼:½;

´« = g =4 µ

whereas the gauge couplings i at the top-quark mass scale are evaluated using i

two-loop RGE,

« ´Ñ µ=58:5½ ¦ ¼:½;

ØÓÔ

« ´Ñ µ=¿¼:½5 ¦ ¼:¼6;

ØÓÔ

½ ·¼:¾97

« ´Ñ µ=9:¿¼ :

ØÓÔ (20)

¿ ¼:¾78 The QCD–QED rescaling factors for the fermion masses are estimated up to 3-loop level

following the procedure of Barger et al [23],

·¼:¼95

= ½:54¼ ; = ½:¼½7 ¦ ¼:¼¼½;

¼:¼87

·¼:8¾9

= ¾:½84 ; = ½:¼¾8 ¦ ¼:¼¼4;

¼:4¿9

·½:85¾

= ¾:488 ; = ½:¼46 ¦ ¼:¼½½;

Ù e

½:859

242 Pramana – J. Phys., Vol. 54, No. 2, February 2000

Seesaw neutrino masses

« ´Å µ « ´Å µ

Z Z where the uncertainties which are calculated from input values of ¿ and ,

are higher than the conventional values [24]. The choice of the initial values of the quartic-

=¼:69 = ¾

Higgs scalar coefﬁcients are taken from the paper by Hill et al [27] as ½ ,

¼:79 = ¼:46 = ¼:6¼ = ¼:¼½

4 5 , ¿ , , . As a result of the numerical solution of the

differential equations (6), (7), we have shown in ﬁgure 1 the evolution of the three Yukawa

´h ;h ;h µ Ø = ÐÒ´=½ µ

couplings ØÓÔ as a function of energy scale GeV , and observed the

Å =¼:59 ¢ ½¼ ØaÒ ¬ =58:77

lepto-quark uniﬁcation scale at Á GeV for the value of .The

Á « Å

i Á values of integrals i and the three gauge couplings at the scale , are numerically

evaluated as

« ´Ø µ=´¼:¼¾¼½; ¼:¼4½8; ¼:¼4¾¾µ i =½;¾;¿; Á

i for

Á =´¼:¼48; ¼:¼46; ¼:¼¿½; ¼:¼6¾µ i = ;b;; : ¾ i for top

Making use of the above values, we present, in table 1, the low-energy values of the coef- C

ﬁcients for all four types (I–IV), while the predictions for the neutrino masses are

;;e

« ´Å µ Z

given in table 2. The uncertainties are calculated only from input values of ¿ and

« ´Å µ Å Á Z , whereas the uncertainties in due to threshold effects and top-quark mass,

are not considered here.

Å = ¼:59 ¢ ½¼

It is important to note that our low value of Á GeV, compared to

those of Freire [18] as well as Parida et al [32], can be understood from the low input

Ñ = ½66:5

value of running top-quark mass, ØÓÔ GeV which corresponds to pole mass of C

175.6 GeV [30]. In table 1, there is enhancement in values for types (II–IV) com- ;e pared to those of type I, and this will show the low value of hierarchical mass ratios predictions. The present numerical predictions from types I–IV seesaw formulae on neutrino masses shown in table 2, could not accommodate both solar and atmospheric neutrino oscillation data, while LSND data can be ﬁtted with the predictions from type I. Attempt

Figure 1. The evolution of the three Yukawa couplings of the third generation as a

= ÐÒ´=½ ¬ =58:77: function of energy scale, Ø GeV) for ﬁxed value of tan

Pramana – J. Phys., Vol. 54, No. 2, February 2000 243

N Nimai Singh and S Biramani Singh C

Table 1. Numerical predictions of the low-energy values of the coefﬁcients deﬁned

Ñ ´Ñ µ i at the respective fermion mass scale i for all four types of seesaw formulae as

explained in the text.

C C C

Type

·¼:¼½7 ·¼:¼¿8 ·¼:¼65

:¿¾4 ¼:¼58 ¼:¼45

I ¼

¼:¼½6 ¼:¼¾9 ¼:¼¿½

:899 ¦ ¼:¼¼5 ¼:7¿4 ¦ ¼:¼¼9 ¼:7¼9 ¦ ¼:¼½8

II ¼

·¼:¼½¿ ·¼:½¼¿

:48¿ ¼:¾¼7 ¦ ¼:¼6¼ ¼:½8¾

III ¼

¼:¼½¾ ¼:¼8¼

·¼:¼89 ·¼:½6¾

:857 ¦ ¼:¼¼¾ ¼:¿48 ¼:¿¼5

IV ¼

¼:¼96 ¼:½¿½ C

Table 2. Numerical predictions of neutrino masses of three families using values i

given in table 1.

Ñ Ñ Ñ

Type (eV) (eV) (eV)

·¼:99 ·¾:75

5 5

:5¾ ¦ ¼:¼8µ ¢ ½¼ ½:54 ½:89 ¢ ½¼

I ´½

¼:77 ½:¿¼

½ ½ 6

:86 ¦ ¼:¼¿µ ¢ ½¼ ´½:¿7 ¦ ¼:¼¾µ ¢ ½¼ ´¿:¼½ ¦ ¼:¼½µ ¢ ½¼

II ´4

·¾:½7 ·½:5¾

5 ¾

¾:68 :¿7 ¦ ¼:¼6µ ¢ ½¼ 7:64 ¢ ½¼

III ´¾

¾:¾4 ½:½8

·¼:¼¼4

¿ ¾

:¼77 ¦ ¼:¼¼¼½µ ¢ ½¼ ´¼:¼½8 ¦ ¼:¼¼5µ ¢ ½¼ ¼:¼¼8

IV ´¼

¼:¼¼¿ to explain all the three data sources (solar, atmospheric, and LSND), or at least both so-

lar and atmospheric neutrino oscillation data, seems to be difﬁcult with our low value of Å

Á . In order to obtain correct magnitude of neutrino masses, our predictions in table 2 Å

can be approximately normalized for higher values of Á [18,20], by a multiplicative fac-

8 ½½

¼:59 ¢ ½¼ =Å Å = ¼:59 ¢ ½¼ Á tor [ GeV Á ]. For an arbitrary value of GeV, there is

enough scope for accommodating experimental data, and in particular, type IV would give

Ñ =´¼:¼77; ¼:¼¼¾; ¼:¼¼76 ¢ ½¼ µ i = ; ; e

eV for , respectively. These can accom- i

modate both atmospheric and solar neutrino oscillation data if the oscillation channels are

! !

e interpreted as and , respectively. In short, the Yukawa couplings uniﬁca- tion in the standard model with two-Higgs doublets, implies an intermediate scale which is

too low to be compatible with any type of the seesaw mechanism. However, with an ad-hoc Å or otherwise motivated use of the higher values of Á , linear seesaw formula with charged lepton masses (type IV) gives correct predictions of the three neutrino masses, and their hierarchical mass ratios compared with recent observational data on neutrino oscillations.

4. Summary and conclusion

We ﬁrst investigate in this work the uniﬁcation of three Yukawa couplings of the third

generation in two-Higgs doublet standard model which might emerge from SO(10) model

¬ =58:77

with a single intermediate symmetry. For a ﬁxed value of ØaÒ , we obtain the

Å =¼:59 ¢ ½¼ uniﬁcation scale at Á GeV. Then the low-energy seesaw formulae with radiative corrections arising from fermion mass renormalization and neutrino-mass-operator coefﬁcients, are obtained. We study both quadratic and linear seesaw formulae with either

244 Pramana – J. Phys., Vol. 54, No. 2, February 2000 Seesaw neutrino masses

up-quark or charged lepton masses, leading to four types (I–IV) of seesaw formulae. Using Å our low value of Á we then evaluate the neutrino masses which are too large in magni-

tude to be compatible with solar and atmospheric oscillation data. However, our numerical

½½

Å =¼:59 ¢ ½¼ predictions for type IV seesaw formula with an arbitrary value Á GeV,

show good overlapping with experimental data of solar and atmospheric neutrino oscilla-

ØaÒ ¬ Å tions. Though we concentrate on single value of and Á in this work, their variations would give wider spectrum of interesting results.

Acknowledgements

One of us (NNS) is thankful to Dr M K Parida for useful discussions.

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