The content of physical massess, mixing angles and CP-violating phases in the
lepton sector of extended standard model, both renormalizable and
non-renormalizable, with arbitrary numbers of the singlet and left-handed
doublet neutrinos is systematically analyzed in the weak basis.Comment: 11 pages, LaTe

Pirogov, Yu. F.

English

arXiv

The content of physical massess, mixing angles and CP-violating phases in thelepton sector of extended standard model, both renormalizable andnon-renormalizable, with arbitrary numbers of the singlet and left-handeddoublet neutrinos is systematically analyzed in the weak basis

Pirogov, Yu F

CERN Document Server

IHEP 00{02Parameter countingfor neutrino mixingYu. F. PirogovInstitute for High Energy Physics,Protvino, RU-142284 Moscow Region, RussiaMoscow Institute of Physics and Technology,Dolgoprudny, Moscow Region, RussiaAbstractThe content of physical masses, mixing angles and CP violating phasesin the lepton sector of extended standard model, both renormalizableand non-renormalizable, with arbitrary numbers of the singlet andleft-handed doublet neutrinos is systematically analysed in the weakbasis.∗E-mail: pirogov@mx.ihep.su1 IntroductionThe mixing of quarks in the minimal Standard Model (SM) of strong andelectroweak interactions is now well understood. It is described by theCabibbo-Kobayashi-Maskawa (CKM) unitary mixing matrix for the chargedcurrents [1], the neutral ones and Yukawa interactions being flavour conserv-ing. As for the lepton sector, the SM exhibits here an extremely simple andeconomic structure. It encounters just three physical parameters, the chargedlepton masses, and predicts no flavour and CP violation. But it has beenwidely recognized that the inclusion of (iso)singlet neutrinos and/or neutrinomasses in the SM would result in the lepton mixing and flavour violation withall the related phenomena such as CP violation, neutrino oscillations [2], etc.(as a recent review see, e.g., ref. [3]).There are two principal dierences between the lepton and quark mix-ings. First, the number of singlet neutrinos relative to that of (iso)doubletones is not restricted by the chiral anomalies and can be arbitrary. Second,the Majorana masses for neutrinos are possible in addition to the Dirac ones.This complicates inevitably the proper SM extensions and proliferates freeparameters. Hence, an immediate problem arises how to extract the physi-cal parameters, to separate masses, mixing angles and CP violating phasesamong them, as well as to conveniently parametrize the mixing matrices.There have been many studies of the related topics. The case with an arbi-trary number of the left-handed doublet neutrinos but without singlet oneswas considered in ref. [4], the one with equal arbitrary numbers of the singletand doublet neutrinos in ref. [5] and a general case with arbitrary numbers ofboth types of neutrinos in ref. [6]. In particular, the last case with only Diracmasses was studied in ref. [7]. Traditionally, all the previous investigationswere carried out by an explicit construction in the mass basis.In the present paper an alternative approach to the lepton parametercounting is adopted. It is formulated in the weak basis entirely throughsymmetry properties of the model before the spontaneous symmetry break-ing [8]. In these terms, all the possible parameter space congurations ofthe SM extended with arbitrary numbers of the singlet and left-handed dou-blet neutrinos are systematically analysed. Both renormalizable and non-2renormalizable extensions of the SM, among them the pure Dirac and pureMajorana cases, are considered. In a consistent fashion, the known resultson the lepton parameter counting for the SM general extensions are recov-ered, and the new ones for the renormalizable extensions are obtained. Therelation between the two countings is claried. The results on parametercounting for neutrino mixing are summarized in tables in what follows.12 Renormalizable extensions(i) Arbitrary case The most general renormalizable SU(2)WU(1)Y in-variant lepton Lagrangian of the SM extended by the right-handed neutrinosreadsL = lLiD= lL + eRiD= eR + Ri@=R−lLYeeR + lLYRC +12CL MR + h.c.: (1)In eq. (1) the lepton doublet lL and singlet eR, R elds mean those in aweak basis where, by denition, the symmetry properties are well stated.It is supposed that the ordinary chiral families of the SM with the doubletleft-handed Weyl neutrinos in number d  3 are added by the singlet Weylneutrinos in number s  0. Let us designate the SM extended in such arenormalizable manner as (d; s)r extension. A priori, one should retain sand d as arbitrary integers, both s  d and s > d being allowed.2 Further,D=  γD is the generic covariant derivative which reduces to the ordinaryone, @= = γ@, for the hypercharge zero singlet neutrinos. Here and inwhat follows the notations CL  (R)C = CR T , etc, are used for the C1In fact, what we are talking about is lepton mixing which is described by a counterpartof the CKM matrix. But one can always choose a weak basis where the mixing matrixof charged leptons is unity. In this sense, lepton mixing is synonimous with the neutrinoone.2We omit the possible vector-like lepton doublets in the present analysis. Hence, withaccount for the most probable exclusion of the fourth heavy chiral family [9], one shouldput in reality d = 3. Nevertheless, we retain d as a free parameter to better elucidate theparameter space structure of the extended SM.3conjugates of chiral fermions. Y e and Y  are the arbitrary complex d  dand ds Yukawa matrices, respectively, and M is a complex symmetric ssmatrix of the Majorana masses for the singlet neutrinos. Finally,  is theHiggs isodoublet and C  i2 is its charge conjugate.Table 1 Parameter counting for the SM renormalizable extension (d; s)rwith d doublet neutrinos and s singlet ones. In this table and the ones whichfollow, the rst and the second groups of moduli for the physical mass matrixMph correspond to the independent mixing angles and masses, respectively(see text).Couplings Moduli Phasesand symmetriesY e, Y  , M d2 + ds + s(s + 1)=2 d2 + ds + s(s + 1)=2G = U(d)2  U(s) −d(d− 1)− s(s− 1)=2 −d(d + 1)− s(s + 1)=2H = I 0 0Mph(d; s)r sd + (d + s) d(s− 1)Mph(n; n)r n2 + 2n n(n− 1)The parameter counting in the weak basis for the lepton sector of the ex-tended SM proceeds as is shown in Table 1. Here G is the global symmetryof the kinetic part of the Lagrangian of eq. (1). Due to the Dirac and Ma-jorana mass terms the symmetry G is explicitly violated so that the residualsymmetry is trivial, H = I.3 The transformations of the broken part G=H(here G=H = G) can be used to absorb the spurious parameters in eq. (1)leaving only the independent physical set, Mph, of them.4 As a result, Mphcontains sd+d+s independent moduli and d(s−1) phases. In this, realMphcorresponds to CP conservation. Stress that only the total number of inde-pendent physical moduli is xed by the weak basis counting. Due to absenceof the left-handed Majorana masses in the weak basis, there are relations in3In what folows, we generally assume that there are no mass textures or accidental massdegeneracy. Otherwise, the residual symmetry would increase, and special considerationof each particular case would be mandatory.4For this reason, parameters corresponding to symmetry G are represented in tableswith a minus sign, whereas those of H with a plus sign.4the (d; s)r extensions between the actual mixing angles and masses. Consid-ering all the masses as independent ones, while a part of mixing angles as afunction of the masses, would result in the formal number of mixing anglesless than their actual number. This may cause some confusion in explicitparametrization. So, it is more instructive to choose all the mixing angels asindependent ones, considering part of the masses as a function of the anglesand the rest of the masses. To decide what is the minimal number of inde-pendent masses, consider the limit M ! 1 corresponding to decoupling ofs heavy Majorana neutrinos. In this limit, the rest of d Majorana neutrinosshould become massless with necessity. Thus d Majorana masses depend ons ones. Clearly, it is impossible to further reduce the number of independentmasses. Finally, of the physical moduli, sd ones are mixing angles, the restbeing masses of d charged leptons and s Majorana neutrinos. At 0 < s  d,there are also s induced Majorana masses, d − s neutrinos still remainingmassless.5 At s > d > 0, all the d + s Majorana neutrinos acquire masses.It is clear that in contrast to the quark sector, the CP violation generallytakes place for more than one singlet neutrino at any d > 0. The last line inTable 1 illustrates the extended n family SM with one right-handed neutrinoper family.6;7In the case (d; 0)r one has G = U(d)2, all the neutrinos are massless andthe residual symmetry increases up to H = U(1)d of the individual leptonnumbers. Hence the number of mixing angles, as well as that of physicalphases, is equal to zero.(ii) Only Dirac masses There is an important case of the SM renor-malizable extension. Namely, the lepton number conservation would forbidthe Majorana mass terms, both the left- and right-handed. In the absenceof these masses the residual symmetry at 0 < s  d would increase up toH = U(1) of the total lepton number. In this case, designate it (d; s)D,2s degenerate in pairs Majorana neutrinos would constitute s massive Diracones, the rest being massless. Hence there would be s(2d− s− 1)=2 mixing5This reflects the fact that in this case the rank of the neutrino mass matrix is 2s.6Stress that in all the tables the number of physical masses, chosen as independentones, is collectively that for both the charged leptons and neutrinos (Majorana or Dirac,depending on context).7Our counting for the renormalizable (d; s)r extension, both at s  d and s > d,disagrees with that in ref. [6] (see remarks in section 4).5angles and s(2d− s− 1)=2− d + 1 phases [7]. It follows, in particular, thatat s = d  n for this reduced type of the (n; n)r extension one would get 2nmasses, n(n− 1)=2 mixing angles and (n− 1)(n− 2)=2 phases in a completeanalogy to the quark sector.The above results are not applicable at s > d > 0. Here the number ofmassive Dirac neutrinos saturates the maximum allowed value d, the rest ofs − d Weyl neutrinos being massless. Hence the residual symmetry wouldincrease up to H = U(s − d)  U(1), so that the number of mixing angleswould be d(d− 1)=2 and the number of phases (d− 1)(d− 2)=2. The resultsare summarized in Table 2 along with two particular cases (n; n − 1)D and(n; n)D with equal numbers of mixing parameters.Table 2 Parameter counting for the SM renormalizable extension (d; s)Dwith only Dirac masses. The number of physical masses is that of Dirac ones.Couplings Moduli Phasesand symmetriesY e, Y  d2 + ds d2 + dsG = U(d)2  U(s) −d(d− 1)− s(s− 1)=2 −d(d + 1)− s(s + 1)=2H = U(1), 0 < s  d 0 1H = U(s− d) U(1) (s− d)(s− d− 1)=2 (s− d)(s− d + 1)=2 + 10 < d < sMph(d; s)D, 0 < s  d s(2d− s− 1)=2 + (d + s) s(2d− s− 1)=2− d + 1Mph(d; s)D, 0 < d < s d(d− 1)=2 + 2d (d− 1)(d− 2)=2Mph(n; n− 1)D , n > 1 n(n− 1)=2 + (2n− 1) (n− 1)(n− 2)=2Mph(n; n)D , n > 0 n(n− 1)=2 + 2n (n− 1)(n− 2)=23 Non-renormalizable extensions(i) Arbitrary case Let us now generalize the preceding considerationsto the most exhaustive Dirac-Majorana case with the left-handed Majoranamasses. The direct Majorana mass term for the doublet neutrinos is excluded6in the minimal SM by the symmetry and renormalizability requirements. Butin the extended SM as a low energy eective theory, it could stem from theSM invariant operator of the fth dimension−L = 12(Cyi)(lCRh i2ilL) + h:c: ; (2)with i, i = 1; 2; 3 being the Pauli matrices, h being a d  d symmetricconstant matrix,   v being the lepton number violating mass scale (sup-posedly of order of the singlet Majorana masses) and v being the Higgsvacuum expectation value. The above operator with the eective isotripleteld i = (1=)(Cyi) reflects the oblique radiative corrections in the lowenergy Lagrangian produced by the physics beyond the SM.8 In the unitarygauge, it yields the following mass and Yukawa term−L = 121 +Hv2CR L + h:c: ; (3)with  = hv2=.There is no nontrivial residual symmetry in this case either, H = I. Asfor free parameters, phenomenological inclusion of such a mass term increasesthe numbers of moduli and phases by d(d + 1)=2 each. Of the extra moduli,d ones are the Majorana neutrino masses, the rest being physical mixingangles. Hence, the extension amounts to d + s independent neutrino masses,d(d+2s−1)=2 physical mixing angles and the same number of phases [6]. Letus designate this general type of the SM extension as (d; s), whether s  dor s > d. The parameter counting for this non-renormalizable extension ofthe SM is summarized in Table 3.A special case without singlet neutrinos, i.e., (d; 0) extension, results ind(d − 1)=2 mixing angles and the same number of phases [4]. Clearly, theCP violation in the lepton sector becomes possible for two or more familieswithout singlet neutrinos at all. On the other hand, the (n; n) extension withn complete families brings in 2n massive Majorana neutrinos with n(3n−1)=2mixing angles and equal number of phases [5]. Hence, CP violation mighttake place here already for one complete family.8Were the isotriplet i be considered as elementary in the framework of renormalizableextensions, it would change only the emerging Yukawa interactions not aecting the massand mixing matrices.7Table 3 Parameter counting for the SM non-renormalizable extension (d; s)with d doublet and s singlet neutrinos. Symmetries are the same as in Ta-ble 1.Couplings Moduli PhasesY e, Y  , M d2 + ds + s(s + 1)=2 d2 + ds + s(s + 1)=2 +d(d + 1)=2 +d(d + 1)=2Mph(d; s) d(d + 2s− 1)=2 + (2d + s) d(d + 2s− 1)=2Mph(n; n) n(3n− 1)=2 + 3n n(3n− 1)=2(ii) Only Majorana masses Let us consider a peculiar case of the gen-eral extension above. In the absence of Yukawa couplings, Y   0, but atnonzero Majorana masses, both left- and right-handed, the residual symme-try is still trivial (H = I) as in the general case. But now the doublet andsinglet neutrino sectors completely disentangle from each other. All the d+sMajorana neutrinos acquire masses, and we end up with d(d − 1)=2 mixingangles and the same number of phases for the doublet neutrinos, without anymixing for the singlet ones, whether s  d or s > d. Let us designate thiscase (d; s)M. The results are collected in Table 4.9 As for doublet neutrinos,this case formally corresponds to (d; 0)M which, in turn, coincides with thegeneral one (d; 0).Table 4 Parameter counting for the SM extension (d; s)M with only Majo-rana masses for neutrinos. Symmetries are the same as in Table 1.Couplings Moduli PhasesY e, , M d2 + d(d + 1)=2 + s(s + 1)=2 d2 + d(d + 1)=2 + s(s + 1)=2Mph(d; s)M d(d− 1)=2 + (2d + s) d(d− 1)=29Stress that the numbers of the physical mixing angles and phases do not depend hereon s. This is because the right-handed neutrinos are sterile in the case at hand, and theirmixing matrix can be chosen to be unity in neglect of any other interactions.84 RemarksWe would like to clarify some discrepancies for the renormalizable (d; s)rextension between our counting in the weak basis and the one in the massbasis [6]. In the mass basis, an explicit new feature of the (d; s)r extension,compared to (d; s) one, is the appearance of the additional symmetry U(d−s),d  s, due to d − s neutrinos being massless. As a result, it is stated inthe cited paper that the mixing matrix for the (d; s)r extension could beobtained from the corresponding general matrix just by deleting in the latter(d − s)2 spurious parameters corresponding to U(d − s). We would like toremark that this procedure though being applicable does not x the numberof independent parameters and generally overestimates the number of theactual ones.To illustrate, note that it would follow from the prescription [6], e.g.,that at d = s  n both (n; n)r and (n; n) extensions would have the samenumbers of mixing angles, as well as phases, n(3n − 1)=2, in addition to 3nmasses. On the other hand, an arbitrary square complex matrix Y can beuniquely written as a unitary matrix times a positive-denite Hermitian one,and a complex symmetric matrix M can be uniquely decomposed in termsof a unitary matrix V and a positive-denite diagonal one, M = V T MdiagV .This means that with account for the global symmetry G we could startin the (n; n)r extension by choosing from the very beginning the Yukawamatrices Y e and Y  as positive-denite Hermitian matrices and M as apositive denite diagonal one. As we have exhausted thus all the symmetryG and there is no nontrivial residual subgroup H , this set of parameters isthe independent physical one. It contains n(n + 1) + n moduli and n(n− 1)phases. This completely agrees with Table 1 and is clearly less compared toref. [6].We trace the origin of the discrepancy between the countings to the con-straint  = 0 in eq. (3). In passing from (d; s) extension to the (d; s)r one, itrestricts d(d+1)=2 phases and the same number of physical moduli. In this, dof the conditions on moduli serve to determine the induced Majorana masses.Altogether, this leaves s independent Majorana masses, sd mixing angles andd(s − 1) phases. At 0 < s  d, there are s induced nonzero masses, d − sneutrinos remaining massless with necessity. As a consequence of the inbornmasslessness for d− s neutrinos, the stated constraint supersedes here thosegained from the U(d − s) symmetry. E.g., according to prescription [6] the9extension (d; 1)r formally corresponds to 2d− 1 mixing angles and d phases,but explicit construction shows that there are actually just d mixing angles,all of them being independent, and no phases at all. Especially clear theabove constraint works at s > d when there appear no massless neutrinosand there is nothing to delete by the related transformations. Nevertheless,the counting of parameters at s > d for the (d; s)r extension proves to be notthe same as for the (d; s) one.5 ConclusionThe parameter counting in the weak basis is complementary to that in themass basis. It allows one to gain clear insight into the independent physi-cal parameter content of the SM extensions, both renormalizable and non-renormalizable, with arbitrary numbers of the singlet and left-handed doubletneutrinos.The author is grateful to V.V. Kabachenko and L.B. Okun for very usefuldiscussions.References[1] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Mas-kawa, Prog. Theor. Phys. 49 (1972) 652; L. Maiani, Phys. Lett. 62B(1976) 183.[2] B. Pontecorvo, ZhETF 34 (1958) 247 [Sov. Phys. JETP 7 (1958) 172];ibid. 53 (1967) 1717 [Sov. Phys. JETP 26 (1968) 984]; Z. Maki, M.Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870; V. Gribovand B. Pontecorvo, Phys. Lett. 28B (1969) 493; S.M. Bilenky and B.Pontecorvo, Phys. Rep. 41C (1978) 225.[3] S.M. Bilenky et al., Summary of the NOW’98 Phenomenology WorkingGroup, hep-ph/9906251.[4] J. Schechter and J.W.F. Valle, Phys. Rev. D23 (1981) 1666; M. Doi,T. Kotani, H. Nishiura, K. Okuda and E. Takasugi, Phys. Lett. 102B(1981) 323; J. Bernabeu and P. Pascual, Nucl. Phys. B228 (1983)21; P.J. O’Donnell and U. Sarkar, Phys. Rev. D52 (1995) 1720, hep-ph/9305338.10[5] I.Yu. Kobzarev, B.V. Martemyanov, L.B. Okun and M.G. Schepkin,Yad. Fiz. 32 (1980) 1590; S.M. Bilenky, J. Hosek and S.V. Petcov, Phys.Lett. 94B (1980) 495; V. Barger, P. Langacker, J.P. Leveille and S. Pak-vasa, Phys. Rev. Lett. 45 (1980) 692.[6] J. Schechter and J.W.F. Valle, Phys. Rev. D22 (1980) 2227.[7] J. Donoghue, Phys. Rev. D18 (1978) 1632; J. Schechter and J.W.F. Val-le, ibid. D21 (1980) 309.[8] A. Santamaria, Phys. Lett. 305B (1993) 90, hep-ph/9302301.[9] Yu.F. Pirogov and O.V. Zenin, Eur. Phys. J. C10 (1999) 629, hep-ph/9808396.11

Parameter counting for neutrino mixing

Abstract

Similar works

Full text

Available Versions

CERN Document Server