Neutrino Masses in Gauge Models

8/22/2019 Neutrino Masses in Gauge Models

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PHYSICAL REVIEW D VOLUME 23, N UMBER 1 1 JAN UAR Y 1981Neutrino masses and mixings in gauge models with spontaneous parity violation

Rabindra N. Mohapatra*Department ofPhysics, The City College of the City Uniuersity ofNehru York, New York, Ne~ York 10031

Goran SenjanovicFermi National Accelerator Laboratory, Batavia, Illinois 60510and Department ofPhysics, Uniuersity ofMaryland, College Park, Maryland 20742(Received 8 August 1980)Unified electroweak gauge theories based on the gauge group SU(2)~ )&SU(2)y,U(1)~ L, in which the breakdownof parity invariance is spontaneous, lead most naturally to a massive neutrino, Assuming the neutrino to be aMajorana particle, we show that smallness of its mass can be understood as a result of the observed maximality ofparity violation in low-energy weak interactions. This result is shown to be independent of the number ofgenerations and unaffected by renormalization effects. Phenomenological consequences of this model at low energiesare studied. Observation of neutrinoless double-P decay will provide a crucial test of this class of models.Implications for rare decays such as p~y, peee, etc. are also noted. It is pointed out that in the realm of neutral-current phenomena, departure from the predictions of the standard model for polarized-electron adron scattering,forward-backward asymmetry in e+e ~p+p, and neutrino interactions has a universal character and may betherefore used as a test of the model.

I. INTRODUCTIONThe nature of weak interactions appears to beintimately connected with the properties of theneutrino. The celebrated P-Q theory' of charged-current weak interactions, which enjoys resound-ing phenomenological success at low energies, wasmotivated on the basis of y, invariance of the Weylequation of a massless neutrino and its generaliza-tion to charged fermions. The standard SU(2)~x U(l) gauge model of weak and electromagneticinteractions' provides a sound mathematical basis

for the (V-A) theory of charged-current weakinteractions and predicts the existence of neutral-current weak interactions which have also beenconf irmed' within present experimental accura-cies. In the standard electroweak model, as inthe current-current p-A. theories, a masslessneutrino and a maximally parity-violating weakLagrangian seem to go hand in hand.In recent years, an alternative approach4 toelectroweak interactions has been proposed ac-cording to which the basic weak Lagrangian isinvariant under space reflections, as are electro-magnetic and strong-interactions. It thereforeinvolves both p-A. as well as &+A. charged cur-rents. The observed predominance of left-handedweak interactions4 at low energies is understoodas a consequence of the fact that vacuum is notsymmetric under space reflection. More pre-cisely, the weak Lagrangian prior to symmetrybreakdown is given byg I, =~2 (JW"+ ZWg),

where Z, ~ = Z~(y, - -y,) and W~ and W~ are the left-and right-handed gauge bosons, respectively. Thenoninvariance of the vacuum under space reflec-tion results in yyzw ypzw and, as a result, allSg Wglow-energy weak processes appear the same as inthe SU(2)~ x U(1) theory, with small corrections[proportional to (@pe~ /pyz~ )'J, undetectable in ex-periments performed to date.In the left-right-symmetric models, ' since bothleft- and right-handed helicities of the neutrinoare included, the neutrino naturally has a mass. 'The important question can then be raised as towhy the neutrino mass is so small. ' There appearsto be a growing conviction among many physiciststhat a satisfactory understanding of small neutrinomass requires the neutrino to be a Majorana par-ticle. ' " This point of view was, in the contextof left-right-symmetric theories, advocated byus in a previous paper, ' where we have shownthat for neutrino being a Majorana particle, onecan obtain the following qualitative relation:The precise form of (1.1) depends, as we shallsee, on the unknown, free parameters of the Lag-rangian. We suggest a class of models where Eq.(1.1) takes naturally an interesting form

m, 'ygp = const x (1.2)relating the mass of the neutrino to the mass ofthe electron. We believe, however, that the im-portance of Eq. (1.1) [and (1.2)J lies not so much inthe precise value of pyzbut rather in the factI65 1981 The American Physical Society


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166 RABINDRA N. 'MOHAPATRA AND GORAN SEN JANOVICthat it provides a deeper physical insight into theconnection between the small neutrino mass andthe maximality of parity violation. In particular,note that in the limit of mz ~, m, - 0 and weakinteractions 6econze pure V-A type. If the recentexperimental results from Irvine" and the SovietUnion" indicating a nonvanishing neutrino massare confirmed, they would provide a support forthe line of reasoning presented above.In this paper, we analyze in detail the questionof small neutrino masses and mixings in the left-right-symmetric models in which Eq. (1.1) holdstrue. We study the naturalness of Eq. (1.1) andextend it to include the neutrinos at higher gener-ations, i.e. , vand v. We also study the impli-cations of our model for various low-energy weakinteractions. In particular, we discuss the pre-dictions for rare decays such as p,ey and )L(,eeeand for neutrinoless double-p decay, "whose ob-servation, we argue, would be a crucial test ofthe model.The rest of this paper is then organized as fol-lows: Section II describes the basic ideas behindthis work, i.e. , emergence of neutrino massesand their natural smallness in the particular SU(2)~x SU(2)zx U(1) gauge theory. In Sec. III we discussthe phenomenology of the model, paying specialattention to the realm .of neutral-current pheno-mena. Section IV deals with the generalization ofthe model to the case of three generations of fer-mions. Section V presents the estimates for rareprocesses that violate lepton number, in particu-lar neutrinoless double-P decay, p,ey, andp,eee decays. It turns out that neutrinoless dou-ble-P decay is the most interesting prediction ofthe model, since its observability requires twomain features of our model to hold true: MBjoranacharacter of the neutrino and reasonably smallvalue for gg~ . Finally, we summarize our workRin Sec. VI. Some of the technical details of thepaper are left for two appendices: in Appendix Awe show how the particular choice of the Higgssector forces neutrinos to be two-component Maj-orana particles. In Appendix Bwe discuss a majoraspect of symmetry breaking in our model; weshow how parity gets broken spontaneously, andmore than that, how the vacuum expectation valuesof left-handed Higgs scalars which give massesto neutrinos are necessarily small (-m~ '). Thatin turn leads to the main conclusion of this paper:in the limit m~ ~, neutrino nzasses vanish natur-ally.II. LEFT-RIGHT SYMMETRY AND SMALL NEUTRINOMASSIn this section, we will derive Eq. (1.1) relatingthe small neutrino mass with the strength of the

V+A charged currents in left-right-symmetricmodels. For the purpose of simplicity, we willwork in an SU(2)~ x SU(2)~ x U(1)~ ~ model. '"Here, we have used the recent observation" thatin contrast with the U(1) generator of the standardmodel, that of the left-right-symmetric modelscan be interpreted as the B-L quantum number.As we show below, this observation provides phys-ical insight"'" into Eq. (1.1). To see this, notethat in left-right-symmetric models, the formulafor the electric charge reads as follows:B-L'Q Isz. + Isa+Since AQ= 0 and if we are above 100200 GeV,b.i~ = 0, Eq. (2.1) leads to

EI~~=24 (B L)-(2.1)

(2.2)This implies that breakdown of parity and breakingof local B-L symmetry are related. Since forthe neutrino B=0, Eq. (2.2) makes it clear why theneutrino ought to be a Majorana particle (sincethen aL w0) and in particular why its mass musthave something to do with gag~ . An explicit reali-Wpzation of this intuitive picture is provided by spec-ifying the Higgs-particle and fermion content ofthe left-right-symmetric model. We illustrateour procedure with one generation of fermions andextend it subsequently to include higher genera-tions.Fermions are assigned to left-right-symmetricrepresentations' of the group as follows:

Q f ll (& 0 &) Q (2.3)To implement the physical picture outlined inEq. (2.2) we will choose a particular set of Higgsmultiplets, ' some of which carry B-L quantumnumbers. The minimal set with this property is(see Appendix B for a detailed discussion of theHiggs sector)

(2.4)Note, incidentally, that the above Higgs multipletshave the same representation content as the follow-ing bilinears in fermionic fields: P =p~g~ or @~Q~and a~, :-g Cr, T,)~ and &,==psCr, v,g~ There-.fore, the conclusions of this paper are likely toremain valid even if symmetry breaking is dynam-ical and there are no elementary Higgs scalars.The various stages of symmetry breaking are


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS WITH. . . 167SU(2)~xSU(2)~x UB ~(1) zz, )~o (~ z 0SU(2)~x U(1).

)=(' 'l (, )=(' 'l(,zzzz 0z (vz Oj

(0 zz'i(2.5)

with g'g in order to suppress g~-Wmixing"and AS = 2 Higgs-particle-induced processes"

At this stage parity as well as local p- L symme-try are broken. The subsequent breakdown ofSU(2)~x U(1) down to U,(1) is achieved via (zt) g0.Switching an (p) g 0 induces a nonzeroz7 "valuefor (s~), but as we show in Appendix B, (~ )=0((zjz)'/y~) (p) (also see Ref. 17).We now proceed to discuss the fermion spectrum, paying special attention to neutrinos. Onecan imagine two, physically distinct situations:(i) There is only one p inthetheory, whose vacuumexpectation value sets the mass scale for bothleft-handed gauge mesons and fermions. One thenattributes tiny fermion mass (mz zzz~ ) to thearbitrarily chosen small Yukawa couptings. Al-though being the simplest alternative, we don' tfind this particularly appealing. As we shall seelater, neutrino mass then tends to be somewhatlarger than acceptable for reasonably light ~~ .(ii) It has been speculated that the small fermionmass may originate from a different mass scalethan the masses of gauge bosons. This is achievedby simply postulating the existence of two P's,with one of them coupling to the fermions and pro-viding their small masses through its small vac-uum expectation value. We would then have gand P~ with (g~)-zzz~/g providing the gauge-mesonmass and (Pz) -mz/ giving the masses to fermions.In this case, we can imagine (Pf) =100 MeV, sothat Pg. need not be much smaller thang. Of course,if such models are right one still would have toexplain why (pz) (zjz~). In this case, however, weobtain more reasonable values for the neutrinomass, on the order of yn, '/m~ .Below we analyze the implications of these twocases on the neutrino-mass question.Case (i) The pa.ttern of symmetry breaking thatfollows from the minimization of the potentialtakes the form (see Appendix B for details)

sector to Sec. III and go directly to fermions, pay-ing of course special attention to neutrinos. Themost general Yukawa couplings are given by&r zzj-zr, 44R+ izA'i44R+3@rpQ~'iz4QikQ~

fg8 QyK +$2 K ~m=h, IC +kg~ z',m=A, K +(24K.

(2.8)Por the v~, vsector, we obtainLl', =, [vJ(vzCvz, + vLC zz*;)+ zzzz(vzzCvzz+ zzz C vg)]

+ (a +,zz') (v~ z R+ v~ v, ) . (2.9)The Eq. (2.9) looks like a mixture of Majorana andDirac mass terms. The situation becomes muchsimpler if we rewrite (2.9) in terms of two-com-ponent spinors z = z~ and N= C(v) r. Uing theproperties of charge-conjugation matrix) C 1

andCyC = -y (2.10)

we easily obtainvz~C v~ = -N CN,vzvi= N Cv= v CN,

so that Eq. (2.9) can be rewritten as2", =,(zzz;v Cv zzzzN CN)+ (,zz+,w')vr CN + H.c.

(2.11)

(2.12)The above expression is a significant simplifica-tion: now all the mass terms are of the Majoranatype; remember that v and N are effectively two-component complex spinors hat is most simplyseen in the representation of Dirac matrices whereWe now summarize the situation with the followingform of mass matrix:

+i(zjz Ct 6 p +p"Cz,h p )+ H. c., (2.7)where g z;p*y, and C is the Dirac charge-conju-gation matrix. This gives rise to the followingmasses for charged fermions:

5g K,K (2.6) 2 ,=(v N )MC +H.c.,N

wherewhere y is the ratio of Higgs-particle self-coup-lings determined from (B13) in Appendix B.We postpone the discussion of the gauge-meson


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RABINDRA N. MOHAPATRA AND GORAN SEN JANOVICand

a h5vt, r b -h, v, c5(h,tc+h, v ). (2.13)

2|"tan 2(= = 2c/b.b-a (2.14)In Appendix A we show that p, and pf, are Majoranaspinors, i.e. , they satisfy the equations that thespinors defined through the abbreviation g'=C(g) r=|jr do." There we also discuss some of the usefulproperties of Majorana fields (the discussion thereis presented in terms of manifest two-componentspinors).We now study the eigenvalues of (2.13), assumingas before &'&, only for the purpose of simplicity.Now, since Qa, c, we obtain

m, = a c'/b, (2.15)mN ~ ~eUsing Eqs. (2.6) and (2.13) one obtains for thelight and heavy Majorana neutrino masses,

( 1 ~h'& ~'h5) vR (2.16)mN hi; Ug.e

We remindthe rea.der that R-m5, /gandvR m& /gAn important feature of the above expression isnoteworthy stressing: in the limit m~ -~ (i.e. ,vR-~), obviously mR -~ and m-0, in whichcase the weak interactions become purely left-handed. That is the main result of our pa.per,promised in the introduction: the V-A limit ofthis theory leads naturally to vanishing neutrinomass, thus providing (at least qualitatively) arational for the smallness of the neutrino mass.Unfortunately, the quantitative character of Eq.(2.16) is definitely less clear: hhand y arefree para, meters of the Lagrangian. Namely, yis an unknown ratio of various Higgs-particle self-couplings tsee Eq. (B.13)], h, is not determinedby the electron mass (m, =h, R in the limit z w'),andh, would only be determined by the value ofTo be specific, let us for simplicity assumeSe 'the natural value y-1 and h, ~ Q,. In this case itis easy to show that the ratio of m, and m ise N~approximately given by

(2.17)

I'he eigenstates of this matrix are therefore givenby v, = v cos & + trt sin (,

Q, =v sin)+& cos $-,with

Now, even if we choose m~ -1 GeV, in order toobtain ~ ~10 eV one requires ~~ ~10~m~ . Thiseis admittedly a rather large value of gyes~ . How-ever, this still leads to interestiv~ predictions forfry ~ oscillations" (i.e. , t-=10' sec, which iswell within the accessible range of present exper-iments) .In conclusion, in this case, for reasonably lightm~ (m5, ~ 3m5,~), the natural value for neutrinomass tends to be quite larger than experimentallyallowed. We should mention, though, that if y issmall (more precisely if yh, '/h5') and if h, -hwe would obtainm-m, /m~, which is definitelyvea reasonable value. For example, if m~ ~ 3m~(a safe lower bound)" we would obtain m, ~ 1 eV.Next we turn to case (ii), which, as we shallsee, predicts naturally more reasonable valuesfor the neutrino mass.Case (ii). This is the case where gauge-bosonand fermion masses originate from different massscales. Now,

h mgz 9e gh m, 'n%.= g m@1~

(2.18)If h/g= 1, substituting again m~ ~ 3m~, leads tothe predictions

m~ ~230 GeV, m, s1 eV.e Ve (2.19)Clearly, this ca,se leads naturally to a small m, .eAlso, the value for m~ is safe concerning thesomewhat stringent bounds coming from the neu-trinoless double-P decay (see Sec. V).Before closing this section, we rewrite the newleft- and right-handed doublets in terms of physi-cal fields (mass eigenstates) v, and N

(~w(4W) ~q0 r/~rwhere R~- m~ /g and y-mz/h (we assume for sim-Lplicity z' z, although now it is not needed since(rtr~)(rjr~)). The fact that in this case p~ doesn' tcouple to the fermions; means that the form ofneutrino mass matrix [Eq. (2.13)] is unchanged,since v~-g'/vR as before (see Appendix B). Themain difference from the previous case is that anatural value for g is now in the 100 MeV region,and will therefore lead to much smaller values forFor example, if we assume h, =h, =h,h ineorder to be specific, we obtain


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS %1TH. . . 169~v N l)

~ (left-handed doublet),(r ight -handed doublet),N, + vl (2.20)

A~= si n8~(W~+W'~) +(c os2 8~)'~ B)Z~= cos L9~g ~-sin g~ tan (9~/ ~

an 8~(cos 28~)"'B, (3.3)where we have assumed lVlajorana condition.N=C(N)r (since N satisfies Majorana equation)and the tiny mixing between p, and &, is given by((= tan $1)(cos 28 )'~'cos 6)~

where tan 8~=g'/(g'+g")'~' ande i h, ab 2 h5v~ (2.21) m '= (K'+K"+4m ')12 eos Og (3.4)It is clear from (2.20) that the right-handed cur-rents are very small until very high energiesP & ppz. Thus, the analysis at low-energy charged-current data" ceases to be useful in determiningbounds on ~~ . We would like to add that in this

model, since the right-handed neutrino is extreme-ly heavy, the astrophysical considerations" donot restrict the mass of the right-handed chargedgauge boson.III. NEUTRAL-CURRENT SIGNALS OF THE MODELSAND CONSTRAINTS ON m g ~ AND mzgWhether our approach can be experimentallydistinguished from the standard model in the nearfuture depends on the right-handed gauge-bosonmasses. In this section, we therefore analyzethe mass spectrum for W~, Z~, , W~, and Z~, the

eigenstates of the gauge-boson mass matricesand remark on the constraints that follow fromthe available neutral-current data. As we men-tioned before, charged-current data does not provehelpful in this regard due to t.he large mass of theheavy neutral leptons (N), which is the right-hand-ed counterpart of p.Below we give the gauge-meson eigenstates andtheir masses. First, in the charged sector,

mz' -2(g'+ g")v~'.In the above expression 8~ has been defined insuch a way that it can be identified with the Wein-berg angle of SU(2)~ x U(1) (hence, the subscriptW) i.e. , e'=g'sin' 8~. Also, note that the cele-brated relation of the standard model '~~ '= pyg~ '2 I Zgx cos 8~ is preserved to the lowest order [compare(3.2) and (3.4)]; it gets corrections of order K'/v' and ~~'/K', but these corrections will be small.Now, let us proceed to analyze the structure ofeffective neutral-current Hamiltonian in this mod-el. We will show that there exists a remarkablefeature of universality of strength in various neu-tral-current processes, which may be used asa test of the model, once the desired accuracy ofexperiments is reached. .To make computations simpler, we employ themethod of Georgi and Weinberg. " Let us brieflyrecall their result. The effective neutral-currentHamiltonian ean be written in the following form:

(3.5)where f, f' stands for any fermions; i, j counts allthe neutral generators but one (arbitrary) corres-ponding to any U(1) subgroup of hn original gaugegr oup~

W, =S"~cos e +gsin e,W, =W~ sin ~+pecos &, (3.1)

2n; =~ C;P,.C; (3.6)with

mtl 2g (K + K + 2@a ).2 (3.2)We shall, in what follows, assume negligibleg~ -W~ mixing(i. e. , e 1 or K' K). In that ap-proximation, g~ and g~ become the eigenstatesof the mass matrix.As for the neutral-gauge-meson sector, we ob-tRill (ill the llllllt K va)

where C,. are defined from the expression for thecharge.q=+C. T. (3.7)In (3.7) IT/ = ITo, T,.'I (T, being left out of compu-tations) .We are now equipped to present our results. Wewill apply the above method to the three distinct, im-portant classes of neutral-current phenomena: neu-trino interactions, parity violation in electron-quarkscattering (applicable to polarized-electron-hadronscattering and parity violation in atoms), and


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IVO RABINDRA N. MOHAPATRA AND GORAN SENJANOVIC

A. Neutrino scatteringIn this case one obtains for the relevant pieceof the neutral-current HamiltonianK = v 2Gpvyp(1+ys)&

x[(1+n)qy" (T,2Q sin'g)p+ (1+P)Py"y,r,y],

where(3.8)

forward-backward. asymmetry in e+e - g+ p, pro-cesses. complicated and it depends on whether case (i) orcase (ii) introduced in Sec. II are realized. Name-ly, in ca,se (i) g -m~ /g and so v~/x again mea, sures the ratio ofm~ /m~. (Recall that v~-x'/vs}.In case (ii), however, x-m&/h and therefore is ex-pected to be of order 1100 MeV and thereforecompletely negligible and so v~ -x'/v~ can betaken practically to be vanishing.Let us now, in passing, describe the situationin terms of heavy right-handed gauge-mesoneigenstates. From (3.4) we have the followingrelation between WR and Z~ masses (we ignorefor simplicity the tiny mixing between W~ and W~):K2 2V+ s~K 2VR K (3.9) cos g~~n '= 2 ~m~z cos20~ (3.12)

(recall that we work in the approximation w'x).B. Parity-violating electron-quark scattering amplitude

Analysis of the neutral-current state includingthese effects have been carried out by severalgroups. Ecker" gives the following bound (using1 standard deviation):Using the same technique, we find that a pieceof the Hamiltonian responsible for parity viola-tion in atoms and in the SLAC experiment on po-larized-electron-hadron scattering can be writtenasXpv = ~ (1+P)[ey(-1+4 sin 0~)eqy" ysTsqey y, eqy" (T,2Q sin'9~)qI.

(3.10)Note that (I+P) denotes the departure from thepredictions of the standard model and is the samefactor as the one accompanying the axial-vectorpiece in neutrino scattering [Eq. (3.8)].

C. Forward-backward asymmetry in e+e ~ p pInthis case, the relevant piece is theeyy, eI(Ly"ysp,four -fermion interaction. A simple calculationgives

m /m -029. (3.13)Taking ma =90 GeV, (3.13) implies m~ ~ 300 GeV.Now, for sin'0~= 0.22, from (3.12) we obtain

m~ = 0.7mz 'Rwhich then gives a lower bound onm~,VR &m ~180 GeVW~or tpl}y ~ 2.2 5&i ~L

(3.14)(3.15)

IV. HIGHER LEPTON GENERATIONSAND LEPTON MIXINGIn Sec. II, we discussed the neutrino mass forone generation only and showed that its small-ness is related to the smallness of V+A charged-current couplings. In this section, we extend thisresult for neutrinos of all three generations anddiscuss the implications of neutrino mixing. Tobegin we denote the leptonic doublets, which areweak eigenstates,

= 2~2 (1+P}eYp YseI Y ys'P '~' (3.11} (4 1)Again, the same departure from the standard-model prediction.It is therefore clear that once the desired ex-perimental accuracy is reached, the above uni-versality of coupling strength may be used as atest of left-right-symmetric models. Let us nowanalyze it slightly more quantitatively. We haveseen that the departures from the standard modelare of the form v~'/x' and v'/vR'. The latter ratiois directly related to the ratio of left-handed andright-handed gauge-meson masses, and so itsmeasure would determine the relevant parameterof the model. The situation v~/x term is more

Working with the same set of Higgs mesons (i.e. ,a~, b,R, and Q), we obtain the following most gen-eral Yukawa coupling allowed by renormalizabil-ity:

Zr sh;;(g;z~saiCgg~+ $;R~sb'RCg;R)=1+ ij iI. jz + ij ii, .~ +H.C. 4.2i~j=1

Notice that nothing would change if there are bothg and P~, as in the case (ii) of Sec. II, since onlyP& couples to the fermions and so (4.2} follows


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS WITH. . . ]71again. Ef we assume w'g and introduce fieldsv, v,.i and N,. =C(v,.)r as before, one obtains thefollowing mass matrix for the (v, , v, v, ) sector:

vs vN (4.3)Mq '.Itt,'gwhere m, M,~, and Slt are 3&3matrices givenby

where

Ij 2 N

(4. j}KImvv)i j p @ij &~R

9 uN)ij xfij &(5itiiw). j= vjih . . (4.4)

( Vjj) sIN) N,N2

(for three generations) .N3

They are related to the weak eigenstates, des-cribed by"e

fvw'}

Np

Again, we see that all three neutrino masses vanishseparately in the limit vs -~ (or m -~). This isthe generalization of the main result of this paperfor higher generations. This, in particular, im-plies that the predominant left-handed nature ofthe leptonic weak currents as well as the corres-ponding hadronic currents at low energies is due tothe smallness of neutrino masses.%'e now give the general method for diagonaliz-ing" Eq. (4.3) and present rough estimates of thevarious mixing angles in our case. To do this,we write the mass eigenstates as

Of course, in our models. w.. Since we ex-~pect X,j and U,.j to be, in general, of order 1 (orat least not small), Eq. (4.6) then implies thatIZ, , I ~0 Jm, )IXijl, IUI d lr,.jl~tt, rm)IU, jl,where', is a typical parameter in the mass ma-trix of charged leptons, expected to be of ordergmorm, ). As a result, in general, we expectvery small mixing between the heavy and the lightMajorana lepton.

V. LEPTON-NUMBER-VIOLATING PROCESSESIn this section we discuss the rare processeswhich do not conserve the lepton number. Wedivide such processes in two categories: pro-cesses which violate the total lepton number andthose which conserve the total lepton number, butviolate electron, muon, or 7 lepton number. Theexample of the first class of processes is theneutrinoless double-p decay (ji+n-p+p+e +e)in which the lepton number is charged by two units,and the second class of processes are the oftendiscussed muon decays: p-ey, p.-3e, etc. forwhich b,L(total) =0. We shall discuss both classes

i.n some detail.A. Neutrinoless double-P decay [{PP) process I

This process can take place in the second orderin the Fermi coupling if neutrinos are Majoranaparticles (see Fig. 1). To see this we write thecharged-weak-current Lagrangian in terms ofphysical lepton fields (for simplicity we firstdiscuss the case of one generation and generalizeit subsequently)(5.1)

as follows

Using Eq. (4.3), we can write(4.5)

WL(WR )Ij (N),'(

WL(WR)

P

e

PmX+MZ =XD,mY+MU = PD~,IX+HZ =ZD. ,MY+'mU = UD~,

(4.6)FIG. 1. The dominant diagrams which lead to neutri-noless double-P decay through exchange of Wi and v;{i=e,p, v) or W& and N;. The cross on a neutrino intern-al line denotes a (Majorana) mass insertion, since it isclearly a term v z(N N ) which can lead to a productionof two electrons in a final state.


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172 RABIN D RA N. MO HAPATRA AN D GORAN SEN JANOVICwhere ment dictates, and m& 100 GeV, Eq. (5.5) thenyields

~ &(3.5)x10-'. (5.6)

(5.3)for m~100GeV. To compare this with experi-ment, we use the analysis of Halprin et al."whoobtain

g ~5x10 ' (5.4)on the basis of the present data in "Ca-~'Ti,Ge- 6Se and Se seKr For illustration, notethat the bound in (5.4) corresponds to the half-life of "Se:T~~g~ ~4x10""years. Clearly then,such effects are extremely small in our modeland in what folIows, we ignore the tiny W~WR (e)and v-N($) mixing.As we shall see next, the situation is quite dif-ferent with the first type of processes which gothrough v' v and N'N, mass insertions. It turns outthat the model predicts amplitudes which oughtto be observable in the next generation of experi-ments. Due to the tiny neutrino masses, the ex-change of heavy right-handed leptons N, (i =e, g, v)will obviously dominate and we discuss it first.It can be shown that in this case q will be givenby"(5.5)Alga pf

where f, is the nuclear factor, estimated byHalprin et al."for A =100 (3P) nuclei to be about0.35QeV. If we take (m~ /m~ )2s ,, as experi-

(5.2}where (=m, /m |see (2.22)] and e is W~Wmix-ing, i.e., lV, = W~ + eW W, = ~W~ +R'~.We first note that in the lowest order the (PP)0process has to go through mass insertions of thetype v'v or N'N or through currents of oppositechirality. From (5.1}and (5.2) it is immediatelyclear that the latter type of contributions areproportional to e$. Now, from the usual P and gdecay, we know that e is small (e & 10 ') in ordernot to conflict with the predominantly left-handedcharacter of such processes. We will presentour estimates in terms of the usual parametri-zation of the (PP)' process, that is, in terms ofthe admixture of left- and right-handed neutrino-electron currents (denoted q hereafter), i.e.,ey&[(1+y,)/2+ q(1-y, )/2]v. We then obtain forthe contribution involving W~W~ and v-N mixing

Detection of this effect would require measuringceoT,~~~ to an accuracy better than 10""years,which can hopefully be reached in the next genera-tion of experiments. In all fairness we shouldadmit that the above estimate depends sensitivelyonrn . However, ifm~ is light, as we believe,' ' wzthen the estimation is fairly good, since N cannotbe much heavier than what we take; otherwiseYukawa couplings would be too large and the per-turbation theory would break down. Namely,using m~ =hvar, m~ = gm~, we obtainm=h/gmRequiring h ~1 leads to the estimatem~~2mw,since g40 GeV, then clearly m~ S480 GeV andwe have a strict bound q ~7x 10"'. In conclusion wehave shown that for low mw our model predictsneutrinoless double-P decay withe =(10 '10 '}.We believe that it makes future experiments evenmore called for.In passing, we comment on the upper bounds onneutrino masses which can follow from the analy-sis of (8P}' decay. Namely, the requirement thatq ~5x 10 can be shown" to lead to the boundg(0,)'m. & 1 kev, (5.7)where 0~ is the Cabibbo-type rotation in left-handed leptonic currents. To see what (5.V) im-plies, let us first recall the laboratory upperlimits onm." rn&35 eV, m &500 keV, andVi ' Ve Vlfm, &250 MeV. We should also mention the cos-mological bounds which result from the require-ment that neutrinos do not dominate the mattercontent of the universe: Zm~50 eV, where theabove limit (quoted somewhat conservatively)applies to stable (r.& 10' sec) neutrinos only.Now one can easily be convinced that v is practi-cally stable. Namely, from m& 500 keV, it' s"Ppossible decay v&- v, +y leads to at least 7, &10"sec. Therefore the cosmological bound app/iesto v& and we conclude thatm&40 eV, in whichcase (PP)0 doesn't provide any new limit onm.However, since v, could be rather heavy (250 VMeV), it can sufficiently fast decay into v,ee, sothat the cosmological bound doesn't apply to it.Therefore, we can conclude that

(5.8)0~)'m&1 keV,which for heavy v, then implies a rather smallmixing angle with the first generation of leptons.

B. Electron- and muon-number-nonconserving processesWe present here simple order-of-magnitude esti-mates of the various muon- and electron-number-


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS WITH. . . 173changing processes: p-e y, p, -3e, muon captureby nucleus, etc.' %'e start the discussion byanalyzing p.-ey decay. The dominant diagramsfor such processes are depicted in Fig. 2. Toset up the notation, we write the most generalform of the amplitude for p. -ey,

m(p. ey-) =e (f +fy, )incv ~q"pewhere m is the muon mass. From (5.9) one de-rives the decay width

(5.9)

rn5&(I -ey}= 6" (lfl'+ Ifsl'}.Our task is then to estimate the leading expres-sions for f and f, We.separate the neutrino andheavy-lepton contributions. By the analogy withprevious models and computations

(5.10)

p(o, ), ,(o, ),,,,m2g (o),,(0,),, (5.12)

where i = e, p,, 7 is the generation index and OL andOR are the Cabibbo-type rotations in the leptonic

&v &Sv 16+2 8 2L (5.11)e g'fv fsN 16 2 6 Rwhere 5and 5are the Glashow-Iliopoulos-Maiani(GIM) factors"

( cos9, , sin9, )&- sin 9~ s cos 9~ R)

(we ignore the tiny mixings between v's and 's,since it doesn't affect the generality of our re-sults). The result stated in (5.12) and (5.13) isthe well-known statement of the GIM mechanism:the amplitude vanishes for vanishing neutral-lep-ton masses or vanishing mixing angles.Using the formula for the usual lepton-number-changing muon decay

2 . 5G~ m".}= 192 ' (5.14)we obtain for the branching ratio B(p -ey) &(gey)/r-(~ evv-.),B(p ey) =B-"(p-ey) +BN(p. -ey)where

+B""(p,-ey), (5.15)B"(p, -ey)=5',

sector introduced in Eq. (5.1). For example, forthe case of only two generationsm 2 m 2V . P~5= sinOL cos8L mw L (5.13)2 2mN ~m5= sinOR cos9Rmw Rsince in this case

WL(W

v( (N)) e(WR)

]u. v)(N) ) eIGL(GR II GL(GR )

B (p, -ey)=~R(y mw 2B""(p ey) = 5 5

(5.16)

NL(v. (N ) e

III'GL (GR)

jul, Vj

GL (GR)

(N() e(WR)

FIG. 2. The leading diagram for. a lepton-flavor-changing process pp. Again, the process goesthrough the exchange of v; and 5& or,nd W&. Inaddition, due to the GIM mechanism, the Goldstone-boson exchanges (denoted by 6& and G& in-obvious nota-tion) are comparable in strength to gauge-boson-media-ted amplitudes, We ignore the physical-Higgs-particleexchanges, by assuming mH mw.

From the limit in (5.2) coming from double-P de-cay it is clear that 5, is desperately small, so thatclearly either B"(p -ey) dominates in the aboveequation or the whole amplitude is negligible.Continuing our assumption that 5~ is reasonablylight, we present some estimates for B(p -ey)= B~(p.-ey}. Its precise value is obviously ob-scured by the lack of knowledge of the GIM factor5even if mw is of order of a few hundred GeV.RTaking for definiteness 5= 10 '-10 ', Eq. (5.16)then givesm 2 10' B(g -ey)= 10 '10 " (5.1V)1 B(p ey) =]0 && 10mw 2 100

Needless to say, varying 5~ would produce dif-ferent values for B(y, -ey}. The above estimate


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174 RABINDRA N. MOHAPATRA AN D GORAN SEN JANOVICcan, therefore, at best be taken as suggestive.We now briefly comment on some other muon-and electron-number -changing proces se s. Theinteresting possible decay model in our modelis obviously p. -eee. This process is suppressedby additional power in the coupling constant, asis clear from a typical graph as depicted in Fig.3. When compared to the p, -ey process, thebranching ratio B{p,-eee) = I'(p, -eee)/I'(p, -evv, )becomes free of any uncertainties and one simplyobtains the order -of-magnitude estimate

AB(u -eee)= ., B(V, -ey).sin 9~ (5.18}

v) (N)) vi (N;)

W, (WR)(WR)

v) {NI) e(WR)

FIG. 3. Some typical diagrams leading to a decaypee. Clearly, there are quite a few more graphsinvolving Goldstone bosons, which we didn' t displayhere.

For sin e~= 0.22, this would give B(p-eee, )/B(p-ey) =(110)%."This is an important predictionof our model: p,-ey and p. -eee are tiedup toeach other and the simultaneous observation ofboth could be used as a crucial test of the ideasdiscussed in this paper. Lacking the hint fromexperiment, we satisfied ourselves by order-of-magnitude estimate given in (5.18). It is clear,however, that a calculation of B(g -eee)/B(p ey)is called for.Other possible muon-number-changing pr ocessesare eg - p, e and muon capture by the nucleus: p,+ g, g)-e + (A,S). It can be readily checked thattheir evaluation is similar to p. -eee and the am-plitude for all three processes are of the sameorder of magnitude.In conclusion, the model we suggest predictsmuon-number-changing processes. In particularwe estimated that B(p, -ey) =10 '-10 " for rea-sonable values ofm: 10&m~ '/m~ 'S100 andR WR(somewhat arbitrary) input 6=10 '-10 '. Otherprocesses such as p. -eee are also possible, with

typical ratios B(p, -eee)/B(p-, ey)=a few /0 (a fea-ture typical of models involving heavy neutralleptons as sources of muon-number-changingdecays). Hopefully, future searches for such de-cays (with improved sensitivity} will be able toserve as a test of this and similar models.

VI. COMMENTS AND CONCLUSIONThe main result of our work, as we emphasizedrepeatedly, is the explicit connection between thesmallness of neutrino mass and the maximality ofparity violation in low-energy weak interaction.Precisely, ue have shown that the neutrino grassis inversely proportional to the mass of the right-handed cha~ged gauge boson R~, zohich means thatthe V- A /imit of left right sy-mmet-ric theoriescorresponds to the vanishing of neutrino mass.

The crucial ingredient which was responsible forour result is the Maj orana character of neutrinos,i.e., the fact that the left-handed and right-handedneutrinos acquire very small and very large Ma-jorana mass, respectively. That in turn is dictatedby the choice of the Higgs sector, which reduces theamount of arbitrariness in the theory. The samechoice ofHiggs multiplets leads to definite pheno-menological predictions in the realm of neutral-cur-rent phenomena and therefore ties the nature of neu-trino states and the value of their masses with theproperties of gauge bosons. We have discussed atlength the experimental implications of our model andconcluded that the most interesting ones, which alsocharacterize the model most uniquely, are the pro-cesses which violate the lepton-number conser-vation. In particular, the neutrinoless double-pdecay (with t I.= 2) turns out to be the definiteprediction of the theory. We found that the rela-tively small m~ (m~~ 3m~ ) leads to appreciablevalues for such amplitudes which ought to be ob-servable in the. next generation of experiments.Our analysis also showed that the strength of neu-trinoless double-P decay is tied up to the strengthof various lepton-flavor-changing processes (withtotal lepton number conserved), such as p.y,p, -eee, and others. Again, it is the mass of W~which affects such processes most and for low R~we predict B(p, -ey) =10 '10 "and B(p, -eee)= (a few%) B(p,ey). The experiments now inpreparation" are likely to be able to observe theamplitudes of such strength and combined withneutrinoless double-p decay can serve as crucialtests of the ideas presented in this paper.Now, what about the actual values for neutrinomasses? As we have discussed at length in Sec.II, the precise quantitative predictions of ourmodel are still lacking at this point, mainly dueto our lack of knowledge of the values of various


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS %ITH. . . 175Yukama couplings and Higgs-particle self-cou-plings. It turned out that typically expected valuesof m, 's depend crucially on whether there is onlyone mass scale in the theory from which bothleft-handed gauge bosons and charged fermionsreceive their mass (in which case the small val-ues of m, , m, m, m&, m, is attributed to verysmall Yukawa couplings) or maybe it is the ex-istence of hierarchy of mass scales which is re-sponsible for rather different values of fermionmasses in different generations and gauge-bosonmasses. " The latter case admittedly requires arather complicated Higgs sector with probablyfour Q's: Q, , Q, Qand P~ with subscripts e,p. , 7, and Wdenoting the fact that they give themass to the first, second, and third generationof fermions, and W~ and Z~ bosons, respectivelyHomever, we find it more appealing on severalgrounds. First, the smallness of first-generationfermion masses is not attributed to arbitrarilychosen small Yukawa couplings, but rather wouldbe related to the smallness of mass scales (Q, ).It is not inconceivable that one may eventuallyconstruct natural hierarchies, in which case thesituation (P,)( P)( Q)( Q~) would emergeas a prediction of the theory, rather than to bepostulated adhac. Also, from the point of view ofour work, this case leads to much more plausiblepredictions for neutrino masses as compared tothe case of single Q, when their natural valuestend to get larger than experimentally allowed.Finally, we should add that in this case the Cab-ibbo-type angles which characterize quark -flavormixings are necessarily small. Take, for exam-ple, the case of two generations and let us forsimplicity, concentrate on d and s quarks only.We then expect the folloming mass matrix:

which leads to the prediction of neutrinolessdouble-P decay. We believe, in view of our re-sults, that the experiments devised to search in-directly for low-mass W~ are even more calledfor.

ACKNOWLEDGMENTSWe thank Tony Kennedy, R. E. Marshak, andRiazuddin for useful discussions. The work ofR. N. M. was supported by a National ScienceFoundation grant and CUNY-PSC-BHE ResearchAward No. RS13406. The work of G. S. was sup-ported by a grant from the National ScienceFoundation.

APPENDIX A: MAJORANA MASS OF NEUTRINOIn this appendix, we briefly recapitulate some

salient features of the theory of a Majorana mass"and also remind the reader how in the limit ofzero-neutrino mass, both a Majorana and Weylneutrino are identical. As is mell known, for agiven four-component spinor g transforming underLorentz transformations as $-8(, where Spv pv (wi'th g&~ = g&and q&~ = (-) &q(-} "4c, there exist two possible Lorentz-invariantbilinears involving the g: (a) g'y, g and (b)grC 'gmhere C is the Dirac charge-conjugation matrix,which satisfies the property CyC '=y. [Inour basis,f 0 - i~,) (0 I ~(io,. 0 f' ' (I 0)'

and (i o

Since m, =(P,), m, =(Q), then clearly the Ca-bibbo angle is bound to be very small (the reason-able value for 6P~ ean be obtained using the sug-gestion which forbids the term dd).In summary, we have shown how left-right-symmetric theories naturally lead to small m ,linking it to the parity violation in nature. Thesemodels agree with the predictions of the standardtheory at low energies, while at the same timepredict small and universal departures in theneutral- current processes. The main character-istic of the model is the Majorana character ofneutrinos (dictated by the proposed Higgs sector)

C = y,y, .] The important difference between thetype (a) and (b) mass terms is that case (a) termis invariant under a phase transformations of g:g-e'"&, whereas case (b) is not. Therefore, fora spin--,' particle without any kind of conservedcharge associated with it, one may choose either(a) or (b) and in particular it is more economicalto choose (b). The reason this is so is that onemay then mork with only two-component spinors.The reduction from four components to two isusually done by requiring the Majorana conditiongc =C(g)r=g in which case (a) and (b) becomeidentical. The above condition yields


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176 RABINDRA N. MOHAPATRA AND GORAN SEN JANOVICwhere Q is a two-component complex spinor. Thefree-particle equation satisfied by @ is"

(o ~ V Bt )Q mo, f*=0,(8 V + at)o, y~+my= 0 . (A2)

X=y, 2 (1/V)' '[Ag(k)Ug(k)e' '"+4 ~ (k)V~(k) e "' "] (A3)

where Aq and A. ~z are annihilation and creationoperators, respectively, satisfying the canonicalantic ommutation relation(&~(k)~ &*,~(k')] = ~~~ ~T. u (A4)

Uz and Vz are two-component spinors which canbe expressed in terms of the orthonormal basis:o, (k), P(k) satisfying the relations: nto. =PtP =1and ntp =0 and

In terms of n and P, we can writeU, (k) =V,(k) = [N(k) ]' 'o. (k)

These two equations have been analyzed in detail inRef. 31 and the Majorana field Q quantized inthis paper. The decomposition in terms of thecreation and annihilation operators can be writtenas follows:

number as a conserved quantity. From this it alsofollows that, for the massive Majorana neutrino,the violation of lepton number is always proportion-al to the factor (m/E, ). This in particular impliesthat"(T(v+ ~ e + ). !/m p! 10o(v+ -e + ) (E] (A7)

In obtaining the above form we had to use the sim-ple equality N~y&8~~ =N which follows fromthe properties of the charge-conjugation matrixstated before: C =C = C ', CyC =;y. SinceM is symmetric and real, it can be diagonalized byao orthogonal. transformation/v ) /'cos] sin()/v, &Pp E-sin( cosg)lN, ) '

The present experimental bound on the ratio ofcorresponding cross sections obtained by Davis"is at the level of 10k.The discussion of this appendix also makes itclear why double p-decay amplitudes (see Sec. V)are also proportional to the lepton mass.Next, we would like to show that v, and N, in-troduced in Sec. II satisfy Majorana equation (A2).%e remind the reader that we have defined v= v~and N C(Ns) r. In turn, that leads us [see (2.13)]to the following Lagrangian for v and 6/:

v 'I2 = vy&a" v+Ny~s "N'+ (vrN r)MC l ! +H.c.Nf(A8)

( m[N(k)] '~'p (k)k,+!k!That in turn diagonalizes the kinetic part of thepotential, so that we get

m[N(k) '~2p(k)k.+ Ik!2= vye" v+ 2m, (vrC v+ v Ct v*)+N ys "N+ ~m(N CN +N C N*) . (A9)

wherem2N(k) '=1+ 0+!k!From Eqs. (A3), (A4), and (A5), it is.clear thatin the limit of m 0, we have

The peculiar factor of ; is just a definition of mand mat the moment; its meaning will be clearfrom the discussion we now present. Obviously8 will satisfy the same equations of motion, sowe will analyze just one of them, say v. Fromv= v~, whereP=+ (I/V)[A, (k) U, (k)e' '"

+&,*(t )V,(k)e-*""] . (A6) !/1 0)=-'(1+r,) =( )It is clear that in this case, particle and antipar-ticle states are different and they restore lepton in our representation, we obtain v= (&), where Qis a two-component spinor. Therefore


13/16

NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS %ITH. . . 1771) ( 0 -zo,-) (0 I)

1 0) ~io& 0 ~ ' (1 Oj ' (0t'0+'m(pt0)~ . ~ ~ ~ ~+H.c. . (A10)

It is then a simple exercise to arrive at the follow-ing form of the free Lagrangian for Q:

with the numbers in brackets denoting SU(2)~,SU(2)z, and U~ z, (1) quantum numbers, respect-ively. We give their transformation propertiesunder SU(2)~ and SU(2), choosing the matrix formfor h~ and 6, (b, =- 1/v 2 w, 6, )ULb LUL, b,R URER UR,

(a2)2&= Qtg&8" Q+ " (pro, Q+ /to, Q*), (A11) Q-U~QU~, Q-Ul QU.

where 0, are the usual Pauli matrices and a4=-ias before. Varying the Lagrangian in Q*, we ob-tain the equation that Q satisfiesTheir charge decomposition is easily shown to be

0'~s "Q =m o2$* . (A12) 1 ~+L~ R

APPENDIX B: THE HIGGS POTENTIALAND THE PATTERN OF SYMMETRY BREAKINGAs we saw in Sec. II, the Higgs sector in ourmodel consists of the following types of multiplets:h~(1, 0, 2), 6(0,1,2),A(2, 2*,0), g =-~.g*~.(2, -'*,o), (a1)

This completes our proof: v, and N, obviouslysatisfy the Majorana equation (A2), i.e. , free-particle equation satisfied by spinors for whichgc =g." We went through the little exercise des-cribed above with an aim to show that we do,notimpose the conditions that neutrinos are Majoranaparticles, but rather that such a condition is theconsequence of the particular Higgs sector and thepattern of symmetry breaking. Namely, choosingtriplets of Higgs scalars hL and hR to break left-right symmetry led automatically to the physics ofneutrinos as described in this paper.

. 1 I, Bgo+

We present first the analysis of symmetry break-ing for the case of single g [i.e. , case (i) of Sec.II]. Towards the end of this section we shall seehow the analysis given below simply carries to thecase of two @'s [case (ii) of Sec. II]: g~ and pz,with Q~ providing the gauge-boson masses and on-ly Q& coupling to the fermions and being respons-ible for their. masses.Now, consistent with the transformation proper-ties of h~, b,; Q and P and left-right symmetry(for simplicity and without losing generality weforbid the trilinear couplings by an appropriatediscrete symmetry)2 2 2V=Q p. ,~trQ; Q~+ Q &q~q, tr(Q; Q~)tr(QqQ, + Q ~'~qitrQ) (j)J Qq Q, p, tr(bi6i+n, b, )i ~ j=j. i~ jsky / =1 ig j,kgl =1

+ p, ((trb ~b ~)'+ (trb, bz)']+p, (trEJdlb ~h~+ tr6sb zbzb, )+pstrhJAIEzhz2 2 2+ Q Qg)trig Q)(trh~bg+trdsbs)+ P))(trb ~6~(j))Q~+trbshsQ; Q))+ ~ y'g(trb~Q;b~Q),iy j=l ie j=l iy j=l (a4)where Q, = Q and Q, =Q. The symmetry of thepotential under parity conjugation (left-right) sym-metry is recovered by the following constraints onthe Higgs-particle couplings (some of them beingequivalent to conditions for hermicity of the po-

Itential):Pi j= I jit ~1212 ~2121& ~ii jk iikj~

IXijkk= A jikkl Ai jkl = Xti jkAkli j= fkki~lj=+y~l j gi P~ ~&I

(a5)


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178 RABINDRA N. MOHAPATRA AND GORAN SEN JANOVIC2

(0 0) (0 0)vz f p~ 0 (B6)(K 0& - (K' 01(0 K'f (0 Kj

The most general form of vacuum expectationvalues of. the above fields consistent with U, (1)electromagnetic invariance is%e shall assume, for simplicity in what follows,that all the vacuum expe ctation values are real(it can always be made true for a proper range offree parameters of the potential)Our aim here is to show the relationship be-tween v~ and v&, without entering the lengthy butotherwise straightforward exercise of proving thatthe extremizing solution is also an absolute min-imum (that was discussed before). We then ob-viously need to discuss the potentiaI as a function,of v~, v&, K, and K' only. We obtain

IV(& &s~ K~ K ) = P(vz+Vs') + (vz +vs )+~ v~(VI. VB ) ( 11 22 P11) K ( 11 22 P22)K (4 12 P12)KK ]+2v~v[(y+y22)KK'+y(K'+K")]+terms which depend on K K only, (B7)

I+ Vz vs + (V2, +V~ )K + pvzvf2K 22 L 2(B9)

with2( 11 22 P11))

P = 2yFrom the extremizing conditions 0 = SV/Svf= sV/sv, we obtain

P vI, =Pvz, +P vzvs + 12'K vf +PK v2 2v~=pv~ + p vgvg + +K vg+PK vg .3 I 2 2 2

(B10)

(Bll)It is a simple exercise [we multiply the firstequation (B11)by V22 and second by v2, and then tosubtract them] to obtain

[(PP')vzv- PK'](vz' vs') = 0 . (B12)It is clear that the possible solutions to (812) are

2 2(a) vz(b) vz 4 vs, in which ca,se

wherep=4(p1+p2), p =2p2 ~

As previously, we will work in the approximationK'K, so that (B7) becomesV(n, ~, &s, K) = 'P(v z+ v') + (v~ +vs')

Ilocal maximum]. That is a solution which weseeke wanted from the beginning parity to bespontaneously broken. The relevance of (B13) isnow manifest, if we writeKVL, = y,va (B14)

$11 iQ Q i(f1

where y =-iI/(p p'). Clearly, when v-", vz -0and so also rn-0 (since m~ vz)Finally, we want to offer some discussion ofcase (ii) discussed in Sec. II. It is a case of twop's; pf and Q, with only 111f coupling to the fer-mions, which enable s us to imagine an interestingsituation: (1t1f)(Q ). We claimed in Sec. II thatone can still arrange that vz, -(1t1f) /v~, i.e. , vIdoes not depend on large mass scale (P). Wenow prove that statement. From (Bl1) and (B13)it is easy to see how to go about it: we shouldforbid the term P~K11, vzvs (if such a term was ab-sent for a single 1t1, we would have obtained vz=0). Now, p11,=2(p~)12 so we need only forbid theterm (y )12trn, z1t1 &Q( tnioecthat we cannotforbid tr&~P~&sf~ term, but it is proportional toK2K1I and so can be made arbitarily small). Letus therefore impose the symmetry &:

pVIVE= I KP P (B13)~f 1t'f' 1t1f ~f '

~a-&a .(B15)

Now, under a choice of the parameters of theLagrangian one can show that solution (b) is aminimum [in that case solution (a) becomes a In this case, the potenti al is going to have thegeneral form (K K K K )


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NEUTRINO MASSES AND MIXINGS IN GAUGE MODELS WITH. . .U(&L, &R, K, KI)

tt (VI, + VR ) (VL VR ) VL VRl 2p vL, =pvt. +p vg vL,

+(&K + OtyK~ )VL + PK VR2 3 I 2p vg=pvR + p v~ vz (B17)

+ 2(VL + VR )("K + O'~K~')+PVLVRK' +(&K + &~KM )VR+ pK VL~

Similarly as before, it is easy to show that2v~v&=yv , (B18)+ terms which depend on ~, v only.

(B16)The main point is that (B11)now becomes

where y= p/(p p') (o'~ term drops out, as ex-pected). That is a. useful result: it justified ourclaim in subsection (ii) of Sec. ff, that mmf /m~, since K-mt/ft, rather than being-m~ '/LPl Q/ ~

*Address through March 31, 1981: Max-Planck Inst. furPhysik und Astrophysik, Fohringer Ring 6, D-8 Mun-chen-40, W. Germany./Present address: Dept. of Physics, Brookhaven Na-tional Laboratory, Upton, N.Y. 11973.R. E.Marshak and E. G. G. Sudarshan, Phys. Rev. 109,1860 (1958); R. Feynman and M. Gell-Mann, ibid. 109,193 (1958).S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Wein-berg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam, inElementary Particle Theory; Relativistic Groups andAnatyticity GVobel Symposium IVo. 8), edited byN. Svartholm (Wiley, New York, 1968).See, for example, C. Baltay, in Proceedings of theXIX International Conference on High Energy Physics,Tokyo, 1978, edited by S.Homa, M. Kawaguchi, andH. Miyazawa (Phys. Soc. of Japan, Tokyo, 1979).J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974);

. R. N. Mohapatra and J. C. Pati, ibid. 11, 566; 11,2558 (1975); G. Senjanovic and R. N. Mohapatra, ibid.12, 1502 (1975).5For a review and references see R. N. Mohapatra, in&eau Frontiers in High-Energy Physics, proceedingsof Qrbis Scientiae 1978, Coral Gables, edited byA. Perlmutter and L. Scott (Plenum, New York, 1978).See also G. Senjanovic, Nucl. Phys. B153, 334 (1979).6For the possibility of vanishing neutrino masses, seeG. C. Branco and G. Senjanovic, Phys. Rev. D 18, 1621(1978); P. Ramond and D. Reiss, Caltech report, 1978(unpublished).The present experimentat. upper limits on neutrinomasses are given as follows: m 35 eV, E. F. Tret-yakov et al. , in Proceedings of the InternationalNeutrino Conference, Aachen, 1976, edited byH. Faissner, H. Reithler, and P. Zerwas (Vieweg,Braunschweig, West Germany, 1977); m&0.56 MeV,M. Daum et al., Phys. Lett. 74B, 126 {1976);andm&250 MeV, J. Kirkby, in Proceedings of the 1979International Symposium on Lepton and Photon Interac-tions at High Energies, Fermilab, edited by T. B.W.Kirk and H. D. I. Abarbanel (Fermilab, Batavia, Illin-ois, 1980).M.Gell-Mann, P.Ramond, and R.Slansky {unpublished).See also, T. Yanagida, KEK lecture notes, 1979 {un-

published).E. Witten, Phys. Lett. B91, 81 (1980); also invitedtalk given at The First Workshop on Grand Unification,1980 [Harvard Report No. HUTP-80-A031 (unpublish-ed)].R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett.44, 912 (1980),F. Reines, H. %. Sobel, and E. Pasierb, Phys. Rev.Lett. 45, 1307 {1980).V. A. Lyubimov et al. , Report No. ITEP-62, 1980 (un-published).For a review of the present experimental situation indouble-P decay, see E. Fiorini, Rev. Nuovo Cimento2, 1 (1971).R. E. Marshak and R. N. Mohapatra, Phys. Lett. B91,222 (1980).R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett.44, 1316 (1980).|8We stress that the present choice of Higgs multipletsis essential if we want to extract the full content oflocal B-L symmetry breaking. In earlier papers,Higgs multiplets of type gL, ={~,0, 1) and gz ={0,i, 1)were chosen [see Refs. 4 and 5 and also R. N. Moha-patra, F. Paige, and D. P. Sidhu, Phys. Rev. D 17,2462 (1978)]. As a result, exact global U(1) symme-tries identifiable with baryon and lepton number werepresent in the theory, thereby making the neutrino aDirac particle. In such cases the connection between{B-L) breaking and parity violation was not apparent.R. Barbieri, D. V. Nanopoulos, and F. Strocchi, CERNreport (unpublished)."%.Maag and C. Wetterich, CERN report, 1980 (unpub-lished).~~M. A. B. Beg, R. V. Budny, R. N. Mohapatra, andA. Sirliri, Phys. Rev. Lett. 38, 1252 {1977);39, 54 (E)(1977) and references therein.G. Senjanovic and P. Senjanovic, Phys. Rev. D 21,3253 (1980).2~G. Ecker, University of Vienna report, 1979 (unpub-lished).22G. Steigman, K. Olive, and D. Schramm, Phys. Rev.Lett. 43, 239 (1979).23H. Georgi arid S. Weinberg, Phys, Rev. D 17, 275(1978). For applications of this method, see G. Senjan-


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180 RABINDRA N. MOHAPATRA AND GORAN SEN JANOVICovic, Ref. 5; G. Costa, M. D'Anna, and P. Marcolungo,Nuovo Cimento 50A, 177 (1979).24As this paper was being prepared for publication, wereceived a paper by V. Barger, P. Langacker, J. Le-veille, and S. Pakvasa fPhys. Rev. Lett. 45, 692 (1980)]which also discusses the mixings of Majorana neutri-nos.5A. Halprin, P. Minkowski, H. Primakoff, and J. P.Rosen, Phys. Rev. D 13, 2567 (1976). See also, R. E.Marshak, R. N. Mohapatra, and Riazuddin, VPI report,1980 (unpublished).26R. Cowsik and J.McLelland, Phys. Rev. Lett. 29, 669(1972).27For earlier work on p,y see, for example, T. P.Cheng and L. F. Li, Phys. Rev. D 16, 1425 (1977);S. Bilenky, S. Petcov, and B. Pontecorvo, Phys. Lett.

67B, 309 (1977); J. D. Bjorken, K. Lane, and S. Wein-berg, Phys. Rev. D 16, 1474 (1977); B. %'. Lee andR. Shrock, ibid. 16, 1444 (1977).J. Iliopoulos, S. L. Glashow, and L. Maiani, Phys.Rev. D 2, 1285 (1970).K. Gotow, invited talk given at the Workshop on WeakInteractions, held at VPI, 1979 (unpublished).H. E. Haber, G. L. Kane, and T. S. Sterling, Nucl.Phys. B161, 493 (1979); P. Ramond and G. G. Ross,Phys. Lett. 81B, 61 (1979).3iFor the detailed analysis of the properties of Majoranaspinors, see K. M. Case, Phys. Rev. 107, 307 (1957).3 R. Davis, in Proceedings of the International Confer-ence on Radioisotopes in Scientific Research, Paris,1957 (Pergamon, London, 1958), Vol. I.

Neutrino Masses in Gauge Models

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