Flavor-changing neutral currents, CP violation, and impure Majorana neutrinos

Dan-Di Wu*

HEP, Prairie View A&M University, Prairie View, Texas 77446-0355~Received 19 February 1997!

Tree level diagonalization of a neutrino mass matrix with both Majorana and Dirac masses is discussed in ageneral context. Flavor-changing neutral currents~FCNC’s! in such models are inevitable. Rephasing-invariantquantities characterizingCP violation in FCNC’s fermion–Higgs-boson interactions are identified. Scenarioswith maximal CP violation are discussed. At the one-loop level, the mass eigenstates become an impureMajorana type, if neutrinos decay. The possibility of a significant change in the originally hierarchical massspectrum for the left-handed neutrinos is explored, with an example of two species of neutrinos. Neutrinooscillations with impure Majorana neutrinos are also discussed.@S0556-2821~97!01715-3#

PACS number~s!: 12.15.Mm, 11.30.Er, 12.15.Ff

Data from more and more neutrino oscillation experi-ments hint at nonzero neutrino masses@1#. Furthermore, theavailable data seem to indicate that the mass squared differ-ences of the neutrinos, if they exist, form a hierarchical spec-trum. This is to be compared with the fact that the masses ofeach type of charged fermions have a hierarchical spectrum.Another reason for the discussion of massive neutrinos isthat they could be a candidate for the dark matter in theuniverse.

Neutrinos are very different from other fermions, becausethey are electrically neutral. Since neutrinos are neutral, theymay have two different types of masses, the Dirac masses,which are similar to the masses of all charged fermions andthe Majorana masses, which require the violation of the fer-mion number conservation. Since fermion number conserva-tion is not regarded as a fundamental principle in most gaugeinteraction models, neutrinos are very likely to have Majo-rana masses, if they are massive.

In the minimal standard model~SM! neutrinos are mass-less. There is no way to write a mass term for the neutrinosin SM without breaking the gauge symmetry of this model.However, one can give neutrinos masses with a minimal ex-tension of the SM. One can, for example, add a Higgs tripletDL to SM and obtain a new term

k i jLc

Li

T CDLcL j1H.c., ~1!

where

cL5S n

L

eL

Dis the lepton doublet. Note, in this equationnL is a Weylspinor, which is the eigenstate of interactions. The Majoranacoupling constantsk i j

L form a 333 matrix. This matrix mustbe symmetric, becausec

Li

T CcL j

5cL j

T CcLi. The antisymmet-

ric part of the coupling matrix, if existent, does not contrib-ute. ~The antisymmetric part is canceled out by itself.! If the

vacuum expectation value ofDL is non-zero, then neutrinoswill be massive, with a symmetric mass matrixML , and thelepton number conservation will be spontaneously broken.One needs both small coupling constants and small vacuumexpectation values~VEV’s! of DL to accommodate for theobserved tiny neutrino masses in such a model.

One can also introduce, instead ofDL , right-handed neu-trinos n

Ri. One then has

gi jnRi

cTCfTcL j

1k i jRmn

Ri

cTCnRj

c 1H.c., ~2!

where the Majorana coupling constantsk i jR form a symmetric

matrix, andm is a mass scale. Here again we meet withMajorana mass terms; the mass terms for the right-handedneutrinos. These mass terms do not vanish, unless fermionnumber conservation is imposed on the model. The Dirac-type Yukawa coupling constantsgi j form an arbitrary matrix.A possible criticism to this model says that since the fieldsnRido not have any gauge interactions of the SM, it does not

look appealing to have them, unless there are some othergauge interactions beyond the SM interactions which involvenRi. After f develops a vacuum expectation value~VEV!,

we obtain a 636 symmetric mass matrix, if there are threegenerations of fermions. The mass matrix of mass terms~2!is, in the form of a 232 block matrix,

S 0 M

MT MRD , ~3!

whereMi j512gi jv andMRi j5k i j

Rm. Whenm50 this is themodel which has only Dirac masses.

Combining mass terms~1! and~2!, a general mass matrixof the neutrinos has the form

SML M

MT MRD , ~4!

which is symmetric@2#, with MLi j5k i jL ^DL&.

Models such as the left-right symmetric models, theSO~10! grand unification models@3#, and their supersymmet-ric extensions, necessarily require massive neutrinos. A char-acteristic property of these models is that they all need neu-

*Electronic address: wu@hp75.pvamu.edu ordanwu@physics.rice.edu

PHYSICAL REVIEW D 1 AUGUST 1997VOLUME 56, NUMBER 3

560556-2821/97/56~3!/1522~6!/$10.00 1522 © 1997 The American Physical Society

trinos to have Majorana masses, if neutrino masses of theorder of the masses of the charged fermions are to beavoided. Indeed, in these models, neutrinos obtain Diracmasses which are ‘‘naturally’’ compatible with the masses ofcharged fermions. In order to produce acceptable tiny massesfor left-handed neutrinos, out of these much too large Diracneutrino masses, a see-saw mechanism must be applied@2#.Consequently, one needs an originally large right-handedMajorana neutrino mass matrix. The effective neutrino massterms in these models can be represented by Eq.~2!, or bythe combination of Eqs.~1! and ~2!. The latter case with anonzero VEV forDL is not popular. In either case, it is notdifficult to make the model satisfy the conditionk i j

L5k i jR by

suitably applying theL-R symmetry of the models.This paper will be divided into two parts. The first part is

limited to the tree-level discussions. The diagonalization of aneutrino mass matrix is discussed in the most general con-text. Flavor changing neutral currents~FCNC’s! interactionsmediated by Higgs bosons are inevitably found, if both Ma-jorana and Dirac masses are present, as they are in mostmodels being discussed. Neutrino relatedCP violations arediscussed by identifying the corresponding rephasing invari-ants. Ways of realizing maximalCP violation are also dis-cussed. First loop effects will be discussed in the secondpart, based on the hierarchical property of the tree level neu-trino spectrum. These effects include impure Majorana statesas mass eigenstates and the asymmetry of decay productsdue toCP violation. The potential of making almost degen-erate left-handed Majorana neutrinos due to a loop effect ispointed out.

First let us discuss the general diagonalization problem inorder to see the FCNC neutrino interactions on a precisebasis. A symmetric matrixS, such as in Eq.~3! or ~4!, can bediagonalized by one unitary matrix in a symmetric way@4#;

UTSU5D, ~5!

whereD is a diagonalized matrix with all elements real andno less than zero, andU5U1

T3eif with U1, a unitary matrixwhich satisfies

U1SS†U1

†5D2,

andf is a real diagonal matrix such as to makeD real anddefinitely non-negative.

If the see-saw mechanism is at work, block diagonaliza-tion of the complete neutrino mass matrix of the type of Eq.~3! with MR@M ends up with a tiny Majorana mass matrixfor the left-handed neutrinos, as expressed in the formula@2#

MNL52MMR21MT, ~6!

whereMR is the large 333 right-handed Majorana neutrinomass matrix andM is the 333 Dirac neutrino mass matrix.Here we assume the original left-handed Majorana mass ma-trix is either zero or negligible (ML50). Otherwise therewill be an additional 333 original left-handed Majoranamass matrix at the right-hand side of the equation. Note thatEq. ~6! is symmetric, which is consistent with its Majoranaproperty. This formula is widely used in the literature, but issometimes wrongly recorded.

Second, one notices that the flavor-changing neutral cur-rents~FCNC’s! in the neutrino sector are inevitable and co-pious when Dirac and Majorana masses are both present inthe models, particularly in any see-saw models of neutrinomasses. Indeed, expressing the unitary matrix which diago-nalizes the full neutrino mass matrix in a 232 block form:

U5S ULN U12

U21 URN D , ~7!

one finds that the following matrices are diagonalized veryaccurately after the full mass diagonalization:

MRD5UR

NTMRURN ~8!

and1

MLD5UL

NTMNLULN , ~9!

whereMNL is defined in Eq.~6!. However the new Majoranacoupling constants between the left-handed neutrinos and theleft-handed tripletDL ,

kN5ULNTkLUL

N5ULNTUR

N*MR

D

VRURN†UL

N , ~10!

are not diagonalized, where left-right symmetry is assumedin the second step. Neither are the new Yukawa couplingconstants among the left-handed neutrinos, right-handed neu-trinos, and the Higgs doubletf0:

gN5ULN†gUR

N . ~11!

Consequently, flavor-changing neutral currents~FCNC’s!mediated by the neutral component of the tripletD0 and bythe neutral component of the doubletf0 exist in general.

One may wonder how these FCNC interactions have af-fected the abundance of different species of relic neutrinosfrom the big-bang universe. A quick examination tells thatthese interactions are too weak to have such an effect. Actu-ally, the average rate of the relevant process~e.g.,nL3

1nL1→DL

0→2nL1) ^svn&Td at the temperatureTd; 10

eV, the assumed heaviest mass of left-handed neutrinos, ismuch smaller than the then Hubble constantH(Td) for allreasonable masses ofDL

0 .As far asCP violation involved in massive neutrino mod-

els is concerned, one notices thatCP violation may comefrom several sources. There can beCP violations mediatedbyWL andWR , which are characterized by their correspond-ing Cabibbo-Kobayashi-Maskawa~CKM! matrices. Of spe-cial interest isCP violation rooted in the nondiagonal neu-trino interactions of Eqs.~10! and ~11!. CP violation inHiggs coupling constants is defined by the imaginary parts ofthe rephasing invariant quartets@5#, e.g.,

D iak 5« i jk«abgk jb

N kkgN k jg

N* kkbN* ~no summation!. ~12!

1Or ULNTMNLUL

N1ULNTMLUL

N if ML in Eq. ~4! is tiny but stillsignificant.

56 1523FLAVOR-CHANGING NEUTRAL CURRENTS,CP . . .

It is found that it is impossible to make a nontrivial quartetby interfering two tree diagrams for the Majorana couplingsin Eq. ~10!. Therefore, tree levelCP violation with FCNCMajorana couplings does not exist. SuchCP violation existsfor the Yukawa couplings in Eq.~11!, which are quite simi-lar to the charged current gauge couplings, except for thelack of universality for the Yukawa couplings. Note thatsince kN ~or gN) here is not unitary, ImD ia

k ~ImD iaN ) for

different processes are different. These quantities are small ifthere is a hierarchy in the matrix elements. Since all thephases in the coupling matrices are subject to change byrephasing the neutrino fields, there are only four independentuseful phases ingN and three inkN, if these matrices are ofdimension 333. Consequently, when oneD ia is purelyimaginary ~so-called having a maximalCP violation!, theothers may not be imaginary. Such a concept of maximalCP violation is widely used in the estimation of baryon ex-cess due toCP violation @6#. The number of pure imaginaryD ia is limited to 3 fork

N and 4 forgN, respectively. One canalso find a matrix with all itsD ia having significant phases;for instance, a matrix with the following phase distribution:

S 0 Ai AiAi 0 AiAi Ai 0

D ,where a zero element means the matrix element is real. Thisis to be compared with maximalCP violation in theCabibbo-Kobayashi-Maskawa~CKM! matrix, where none ofthe quartets can be made purely imaginary because of theunitarity constraint.

Other interesting newCP violation sources are in thecharged Majorana couplings. For example, theDL

2 couplingof Eq. ~1!,

lLi

T Ci~UL

lTkLULN! jDL

1nL j

1transposed1H.c. ~13!

There will be a similar term for the right-handed neutrinos, ifthe model isL-R symmetric and there is only one pair ofDL - DR :

l RiT C

iSUR

lTURN*

MRD

VRDj

DR1n

Rj1transposed1 H.c. ~14!

Of course, each of these coupling constants have their owncorresponding quartets to be discussed.

The above discussions are only valid at the tree level.When loop effects are introduced, there will be some moreinteresting physics, in particular, the physics somehow re-sembles that in theK02K0 system. This piece of physics,especially that of right-handed neutrinos, has recentlyaroused some enthusiasm due to a brilliant paper by Flanz,Paschos, Sarkar, and Weiss~FPSW! @7#. Essentially, theWeyl fieldsn

Randn

R

c , which are the eigenstates of interac-

tions, are mutuallyCPT conjugated. This system wouldhave been an exact copy of theK02K0 system with asym-metric decay products~e.g., thel1/ l2 ratio is not 1!, if mix-ing mass terms betweenn

Rand n

R

c were not forbidden byLorentz invariance. FPSW found a two species system,

which may fulfill that kind of mixing. However, their discus-sion is limited to the scenario in which the tree level massesof the two species are almost degenerate. A further study ofsuch systems, with hierarchical tree level masses to startwith, should be very interesting. An extension of the discus-sion to the left-handed neutrinos, which are the focus of thepresent neutrino experiments, will in particular be attempted.

Consider two neutrino species of the same chirality, theHamiltonian at the tree level after mass diagonalization is(n

R[n, and assumingCPT)

~n1c ,n2

c ,n1 ,n2!H0S n1

n2

n1c

n2c

D , ~15!

where

H ~0!5S 0 0 M11 0

0 0 0 M22

M11* 0 0 0

0 M22* 0 0

D . ~16!

Note here only neutrinos with a specific chirality~e.g., nR

andnR

c , instead ofnL

c) are considered. This form is conve-nient for separating the absorptive part from the dispersivepart of the Hamiltonian, which will become clear in a mo-ment. At the tree level, the Weyl and Majorana states areequivalent, so far as the mass~and the decay lifetime! eigen-states are concerned. This is not true when loop effects areconsidered. The loop corrections~see Fig. 1! to the zeros inH(0) are convergent@8#, if the theory is renormalizable andthese corrections do not have counterterms. Including loopeffects, the total Hamiltonian is expressed as

FIG. 1. A one-loop contribution to the mass matrix.

1524 56DAN-DI WU

H5H ~0!1H ~corr!5S 0 0 a11 a12

0 0 a12 a22

a11 a12 0 0

a12 a22 0 0

D 5S 0 0 M112i

2G11 M122

i

2G12

0 0 M122i

2G12 M222

i

2G22

M11* 2i

2G11* M12* 2

i

2G12* 0 0

M12* 2i

2G12* M22* 2

i

2G22* 0 0

D . ~17!

There are no odd terms of the eigenvaluel in the eigenequa-tion for H (0)1H (corr). One therefore has

l252l1 ,l452l3~ Iml1<0,Iml3<0!. ~18!

The eigenvectors turn out to be almost Majorana states, orimpure Majorana states as they are called. These eigenstatesMb5(xb ,yb ,zb ,wb), with b51,2,3,4, are expressed as, upto normalization constants,

xb5~a11a222a122 !a121lb

2a12,

yb52~a11a222a122 !a111lb

2a22,

zb5lb~a22a121a12a11!,

wb52lb~a11a111a12a12!1lb3 . ~19!

A phase redefinition of the eigenstates withb52,4, whichare almostCP odd, will change the signs of their eigenval-ues, so as to let them have positive widths. In other words,M1 and M2 are two orthogonal mass eigenstates with anexactly equal mass and width. In the following we will takea phase convention to makea115a11, a225a22.

It is easy to discuss the solutions in two special cases.Case 1: ua22u@ua11u@ua12u. One finds l1'a22,

l3'a11 and to the leading orders (l3 must be calculated tothe next leading orders in order to obtain the following an-swer! the eigenvectors are (d5a12/a22, d 5a12/a22,g5a12/a12)

M1;~d 1 d 1!,

M2;~2d 21 d 1!,

M3;~1 2gd 1 2d!,

M4;~21 gd 1 2d!. ~20!

Note that the mass~decay! eigenstates are of impure Majo-rana type. Assuming charged leptons inn i decay while anequal amount of anticharged leptons inn i

c decay, one findsthat the lepton-antilepton asymmetry in the decays of theseimpure Majorana particle is

d15G~M1→ l21x!2G~M1→ l11 x !

G~M1→ l21x!1G~M1→ l11 x !

5ua12u22ua12u2

2ua22u25

ImM12G12*

M222 1G22

2 /4,

d35G~M3→ l21x!2G1~M3→ l11 x !

G~M3→ l21x!1G1~M3→ l11 x !5d1 . ~21!

The formula ofd1 can be further expressed by nontrivialD ia as illustrated in 1986 by Ref.@5#.

Case 2:ua12u@ua22u@ua11u. In this case, theG part of theHamiltonian is negligible. The interesting new physics istwofold. First, the masses are enhanced froma11 anda22 toabout a12. Second, the splitting is enhanced froma22 toAa22a12. Since both masses are now of the order ofa12, thisis a fascinating mechanism to obtain almost degeneratemasses~in terms of their mass ratio being close to 1! and alarge mixing. It seems that this mass spectrum is not favoredby the present data, although the present data are still to beclarified. The mass eigenstates are now

S M1

M2

M3

M4

D '1

2S 1 1 1 1

1 1 21 21

21 1 1 21

21 1 21 1

D S n1

n2

n1c

n2c

D , ~22!

which are pure Majorana states if the decay rates are ne-glected. Looking at the 434 full mixing, one may wonderwhether it is necessary to work on a 434 mixing matrixwhen discussing neutrino oscillations of two species of neu-trinos. To discuss this, let the gauge couplings be in the434 form

~ l L1W1 l L2W

1 l L1c W2 l L2

c W2!VS M1

M2

M3

M4

D , ~23!

where the 434 mixing matrixV is

56 1525FLAVOR-CHANGING NEUTRAL CURRENTS,CP . . .

V51

2S r1 r1 2r2 2r2

r2 r2 r1 r1

r1 2r1 r2 2r2

r2 2r2 2r1 r1

D ,~r65cosu6sinu!, ~24!

where u is the tree level Cabbibo angle between the twospecies. Suppose at the production point a neutrino is pro-duced together withl 1

1 , then its wave function at a later timet will be

c~ t,n1!5 12 @~r1M11r1M2!e

2 im1t

2~r2M31r2M4!e2 im3t#

5 12 @r1~n11n2!e

2 im1t1r2~n12n2!e2 im3t#.

~25!

One then has, at the detector,

u^c~ t,n1!u l 1,2&u2512 $16cos2ucos@~m12m3!t#%. ~26!

We find this formula very different. A special situation iswhenDmDt@2p, whereDt is the uncertainty of the timemeasurement. In this situation the oscillation part is wipedout and the two species seem to be 45° mixed.

Finally, let us calculate the loop diagrams in Fig. 1 inorder to estimate the size of the effects. The boson and fer-mion in the loops of Fig. 1 can be (D2/D0,l1/n), or(f1/f0,l2/n), and the first combination has a potential tocontribute a significant effect. It has been realized that theHiggs-fermion couplings may be large, since the discoveryof the top quark. As an example, the first combination will beconsidered here. For right-handed neutrinos, in the basiswhere the right-handed neutrino masses are already diago-nalized at the tree level,

a125mR1mR2~mR11mR2!

VR2 (

iM i

RI i, ~27!

where M iR5(U1i

l U2il* )

R, I iR5(1/16p2)(ml i

2/mR2)

@212 ip(ml i2/mR2

2 )] andmRis the mass ofDR . The calcu-

lation is scale (P2) sensitive because the outside propagatoris (P” 1m

R2)/(P22mR2

2 ). The uncertainty in the momentum

flow (P2) will disappear for special physical processes.P2 ischosen to beP25 1

2(mR12 1mR2

2 ) in Eq. ~27! and the assump-tion of P2@mR

21ml2 is made.

The diagrams in Fig. 1 for left-handed neutrinos do notenjoy Glashow-Iliopoulos-Maiani~GIM! suppression, andtherefore are divergent. One may introduce nondiagonal ki-netic terms as the counter terms, once the number of counterterms needed is finite.~Or the counter terms introduced areof dimension 4 or less.! A more careful calculation is neededin order to obtain loop corrections. A guess is that because ofthe smallness of the left-handed neutrino masses, which areproduced by the see-saw mechanism, the corrections can belarge to satisfy the condition for case 2, perhaps for one pairof the left-handed neutrinos. This scenario can particularlyhappen in models with both Dirac and Majorana masses forcharged particles@9#. In these models, opposite charged lep-tons coexist.

The FCNC and Majorana interactions among neutrinosmay play a role in the neutrino scattering in the early uni-verse when the temperature is very high. These interactionsdo not respect lepton number conservation, e.g., one mayhave

nR1n

R→DR

0→ nR1 n

R.

This process is possible becauseDR , which is a componentof the right-handed triplet, develops a VEV.DR is part of the126-plet in the SO~10! models ~in some models, 126 is acomposite field!. These interactions provide a vehicle forlepton and antilepton numbers to reach an equilibrium atextremely high temperatures, even if there is a large leptonnumber excess at the beginning. On the other hand, the ex-istence ofCP violation in the neutrino sector plus leptonnumber nonconservation processes may contribute to abaryon number excess immediately after the decoupling ofsome heavy particles@6#.

In conclusion, massive neutrino models are likely to haveflavor-changing neutral currents in the Higgs boson mediatedneutrino interactions. In the models that use the see-sawmechanism to explain the smallness of the neutrino masses,FCNC’s are inevitable.

The author acknowledges useful discussions withR. Arnowitt and Z. Z. Xing. He thanks X. M. Zhang, H. Q.Zheng, and D. Lichtenberg for encouragement. This workbegan during a summer visit to Texas A&M University. TheNational Science Foundation partially supported this workby a NSF HRD grant.

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56 1527FLAVOR-CHANGING NEUTRAL CURRENTS,CP . . .