on spontaneous cp violation in the lepton sector

Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

O N S P O N T A N E O U S CP V I O L A T I O N IN T H E L E P T O N S E C T O R

W. G R I M U S Institut J~r Theoretische Physik der Universitiit Wien, A-1090 Vienna, Austria

and

H. N E U F E L D CERN, CII- 1211 Geneva 23, Switzerland

Received 15 December 1989

After a general discussion of CP transformations in the lepton sector we apply a class of non-standard CP symmetries to the Zee model. We show that the resulting cases are all equivalent and give rise to a Zeldovich-Konopinski-Mahmoud lepton number and to neutral flavour conservation. The mass, the magnetic moment and the electric dipole moment of the corresponding Dirac neutrino are calculated.

The origin of CP violat ion is still an open problem in part icle physics. In the lepton sector of the standard model [ 1 ] there is nei ther mixing nor CP violat ion because the neutr inos are all massless. How- ever, in most extensions o f the s tandard model neutr inos become massive and therefore also the quest ion o f CP violat ion or conservat ion in the lepton sector arises. Here, we want to apply general CP t ransformat ions [ 2 ] to this problem. We will concen- trate on a certain modificat ion of the s tandard model, originally proposed by Zee [3 ]. Many o f our consid- erat ions are, however, easily transferable to other cases. We construct a three-generat ion model where non-tr ivial CP invar iance induces a Ze ldov ich - K o n o p i n s k i - M a h m o u d ( Z K M ) type lepton number [4] and neutral f lavour conservat ion ( N F C ) in the Higgs sector. We discuss the propert ies o f the emerg - ing Dirac neutr ino.

First , we want to formulate the cri teria that a given lagrangian is CP invariant . As gauge group we choose SU (2) × U ( 1 ), the leptons are introduced as the usual lef t-handed doublets and r ight-handed S U ( 2 ) singlets:

(eL) Li= [z ,' lgi, l<~i<~n~. (1)

The scalar sector contains a certain number n , of Higgs doublets ~,~ and also n, S U ( 2 ) singlet fields qa + with charge + 1:

~ = ~ e o ) , l<~ot<~n~, ~1 +, l<~a<~n,1. (2)

The condi t ion for CP invar iance o f a lagrangian .~ is now that one can find uni tary matrices VL,n(n~ di- mens iona l ) , V~(n~ dimensional and V.(n , dimensional such that ~ is invar iant under the tansforma- tion [2,5,6]

L ( x ) ~ VLCL*(Y~), l t c ( x )~ VRCI*R(YC),

¢,(x)~V,~¢,*(.~), ~+(x)-~v.~-(~),

W u ( x ) o ' ~ ( - l )~°uW~,(.~)o'T,

B u ( x ) - , ( - l )~°~Bu(.~), (3 )

with 2~= (x °, - x ) . The CP matrices, VL,R, V¢,, V, I account for the fact that the weak eigenfields are indis- t inguishable prior to spontaneous symmetry breaking.

The CP t ransformat ion (3 ) is, o f course, basis de- pendent. Unde r a basis t ransformat ion

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 521


L=L~L', IR=URI'R, q~=l.,%q:",

the CP matrices transform as

v;=u~v~uL v'~=u~v,,v*~,

q + = b~q+ ' ,

(4)

(5)

This freedom to choose a basis can be utilized to achieve certain simple standard forms [ 7 ] for the CP matrices VL,R, V~, V,: (O)

02 VCp ~ "•.

0,. 1

( cos~9 i s inOi '] o, \ - s i n O i cosOiJ '

O<O,<~n. (6)

Of course, the submatrices O, cannot be diagonalized by transformations of the type (5). The standard form (6) shows that the CP transformation (3) is in general not equivalent to the trivial one with all matrices Vr.n, V¢~ V, being unit matrices.

The kinetic terms together with the gauge interac- tions are automatically invariant under the general CP transformation (3 ) , whereas the Yukawa terms are subject to certain constraints. The Yukawa inter- action of leptons and scalars is of the general form

rt~ rtr/

- ~ = ~ £r.,~,,l,,+ E LTC-'S,~q+i~zL+h.c. o t = l a = l

(7)

Because of C, ° invariance the no-dimensional coupling matrices F~ and A~=-A~ v have to fulfil the conditions

r~= E vIr#vn(v~)/~,, (8) #= !

and

nq

~ = Z V~A~VL(V,)b~, (9) b = l

respectively. There are also constraints for the pa-

rameters of the Higgs potential which are, however, not important for our purpose.

The gauge group SU(2) × U ( 1 ) is broken down to U(1)cm by the scalar vacuum expectation values < q)o > vac = v~, After spontaneous symmetry breaking the mass matrix of the charged leptons is given by

nG

M~= Z v.r~. (10) C~=I

The weak eigenfields L, 1R are related to the mass eigenfields £,/'R by the unitary transformations

L = U'L£, ln=Un[R. (11)

The matrices UL.R are determined by the require- ment that

Mz= U~ A/[tUR (12)

is diagonal and positive definite. To get a better understanding of the nature of non-

standard CP transformations let us discuss the consequences of strict CP conservation in the lepton sector. Therefore wc assume for the time being that also the vacuum conserves CP, i.e.

rt~

v,,= ~ (V~)~av ~. (13) B = I

Combining eqs. (8), (10), ( 12 ) and ( 13 ) yields

where 19L = U~. VL U{ and I?R = U~ VR U~ are the lepton CP matrices in the mass eigenbasis. From eq. (14) we get

[ ~,., ~ 1 = [ ¢'R, , f # ] = 0 . (15)

Our crucial assumption (obviously,supported by the observed lepton mass hierarchy) is now the non- degeneracy of _~/~. In this case eq. ( 15 ) implies that 17"L and I?R are diagonal, which means that V:= Vn=~ in the standard basis defined by (6). The CP invariance condition (8) is now reduced to

t t~

F*~= ~ Fa(V~)~. (16) p = l

Eqs. (13) and (16) imply v ~ = 0 , / ' , = 0 for I ~<o~<2m and v,~= v*, F,~ = F * for 2 m + 1 ~< a~< n~, if V~ has the form (6).

In other words, in the case of strict CP invariance

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and a non-degenerate lepton mass matrix the CP transformations ofleptons can only be realized in the trivial way (Vt.,R='fl). The same is true for those scalars which are coupled to charged leptons.

On the other hand, if the CP transformations are realized in a non-trivial way (e.g. VL,R#~), CP invariance must be broken spontaneously, because oth- erwise the mass matrix of the charged leptonS would be (pairwise) degenerate. So, in models with a non- trivial realization of CP invariance, non-degeneracy of the lepton masses is a manifestation of spontaneous CP violation [ 6 ].

Let us now investigate such a model with non-trivial CP invariance in some detail. We consider a model with three lepton generations (no=3) , two Higgs doublets (n~= 2) and one SU (2)-singlet scalar 1/+ (n ,= 1 ). For the CP matrices we assume

VL=VR=V(O):= - s i n O cosO , 0 0

0<O~<½n, V~='D, V~=I . (17)

Note that the choice t~:~ VR would again be in con- flict with the non-degeneracy of Mr. The CP invariance conditions (8), (9) determine the form of the Yukawa matrices F, , d. In the case O= ½n we obtain

C,= - b * a* 0 , a,~,b,~eC, c , : ~ , (18) 0 0 c,

A=id -1 0 ; deN. (19) 0 0

For 0 < O < ½ ~, F~ and A have the same structure, but with real parameters a~, b,. In both cases, F, and F2 are simultaneosly diagonalizable by a biunitary transformation [ 8 ]. Using this fact we can choose a basis where F1 and/"2 are diagonal

Fl=diag(gi,gl,c,), g len +, ClsN,

F2=diag(g2 eiLgz e-i~,c2), g~e~ +, c2eg~. (20)

The coupling matrix 3 remains unchanged. Although CP transformations with different O are

not equivalent, they all lead to the same Yukawa couplings (19), (20) for the model under consideration.

It is therefore obvious in the basis where F~ ( a = 1, 2) is real that the model is invariant under a horizontal SO(2) ={ V(O)10~< O< 2n}

L-,V(O)L, IR-~V(O)I~,

,~.--,q,,~, ~ + ~ + . (21)

The equivalence of all cases with 0< O~< ½~ is, however, not a characteristic feature of theories with general CP invariance, but rather a consequence of our representation content. If we had introduced right- handed neutrino singlets vR the obvious extension of (17) would lead to two different models for 0 < O< ~ and O= ~ , respectively. In the first case one would obtain a model with a diagonal lepton mixing matrix but in the model defined by O= ½~ mixing would be non-trivial [ 5 ].

The couplings (20) of neutral scalars to leptons are flavour-diagonal. We observe here the remarkable fact that a CP transformation of the form (17) enforces neutral flavour conservation in a non-trivial way [5,6].

In addition, the Yukawa couplings ( 19 ), (20) are invariant under a global U ( 1 ) × U ( 1 ) symmetry.

LI ~ e i'~ LI, lm ~ e i~'/'m ,

L2--*e-iC~L2, lRz~e -i~ IR2,

L3 --,ei,~ L3, lR3 ~ei#/R3 ,

@.--+ @~, r/+ -~r/+ , (22)

which means that CP invariance has also induced a ZKM lepton number [4] L1-L2 and, separately, a lepton number L3. In the basis with real F,~ the ZKM lepton number corresponds to the above mentioned horizontal SO(2).

Let us now turn to spontaneous symmetry breaking. The lepton numbers as well as NFC remain conserved. A glance at eq. (20) confirms our general considerations about the connection between a CP invariant ground state and the degeneracy of the lepton mass matrix. A CP violating vacuum is thus nec- essary in order to reproduce the observed lepton mass spectrum.

By a suitable orthogonal rotation of the Higgs dou- blet fields and a subsequent U ( 1 ) gauge transformation one can always obtain

(q)°)vac=v~>~0, (q)°)vac=iv2, v2>~0, (23)

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without changing the structure of the Yukawa couplings. In this basis, spontaneous CP violation corresponds to v2 ~ O.

The masses of the charged leptons are now given by

m t = [vlgl +iv2g2 ei~l ,

m 2 = [vlgl + i v 2 g 2 e - i ¢ l ,

m3= Ivlcl +ivzc2[ . (24)

The observed lepton mass hierarchy strongly con- strains the phase ~p..(ei~-i) and also requires

1 --2 vtgt~_v2g2~-2x/ml +m 2. To see this, we add and subtract, respectively, the squares of the first two equations in (24), obtaining

( V l g l ) z + ( V z g 2 ) 2 = l 2 2 2 ( m , + m 2 ) , ( 2 5 )

and

4(vtgt ) (v2g2) sin rp= m 2 - m2t. (26)

Eq. (25) can be identically fulfilled by writing

v,g, =~/½ (mlZ +m~) cosy,

x 2 5- (27) v2g2 =x/2 (rnl +m2) sin y,

where v~g~ >10 and v2gz>~O implies cos 7>/0, sin y~>0. Inserting eq. (27) in eq. (26) yields

2m~ (28) sin 2ysin ~o= 1 m~+m~ "

We assume m~ << rn2 without loss of generality. Eq. (28) shows indeed that sin 27~- 1 (~cos y~- l/w/-2) and sin ~o--~ 1 ( ~ e i~ = i ). It is thus convenient to intro- duce the small angles 3= i n - 27 and ~ = ½ n - ~. Writ- ten in terms of these parameters, eq. (28) reads approximately

82+~2~ - 4m~ (29) m~ +m~ "

The phenomenology of the Zee models has already been discussed extensively in the literature [ 3,9 ]. We will therefore restrict ourselves to those aspects of our model which are characteristic consequences of CP invariance. With the convention rn~ <<m2 we have three possibilities to relate l,, 12,/3 to the physical leptons e,/1, z:

l~=e, lz=z, 13=12 (Le-L.~,L, , ) , (30b)

l l=# , /2=r, 13=e ( L u - L , L , . ) , (30c)

where wc have also indicated the conserved lepton numbers L~ - L 2 and L3.

A possible (higher-order) neutrino mass term

12v~ C-tM,,vL +h.c. (31)

is, of course, restricted by the conservation of these lepton numbers:

( ~ my i ) M~= ~ 0 . (32)

0

The mass matrix (32) corresponds to one massive Dirae neutrino Up= Yr., + (VL2) c with mass I m~l and a massless state vL3. This radiative neutrino mass m, is triggered by a trilinear term in the Higgs potential

2q + rb] q32 + h.c. (33)

with q3z = irrz r.b~ and a coupling parameter), which is real because of C1' invariance. After spontaneous symmetry breaking this term induces a qO + -r/+ tran- sition. The original charged Higgs fields q~ ~-, q~J', r/+ are related to the charged would-be Goldstone boson G ÷ and the physical mass eigenfields II;-, H + by the transformation

iva,v o) vl,v o

q+ \ O 0

(i ° X COSV sin q/ | H ~ J (34) - s i n V cos ~ , / \ H ~ /

with v2=v~ +v~ = (2x/~ GF)-L The masses M~,2 of fI~2, the mixing angle u/and the coupling constant L are related by

22v=sin 2g (MI 2 - M ~ ) . (35)

The explicit form of the neutrino mass m~ up to first order in d, e, produced by one-loop II~.2 ex- change, is given by

l~=e, 12=#, 13 =~" (L~-Lu , L~), (30a)

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dvsin 2~' [ v~ - v 2 M~ m._~ 32~2vlv2 L v 2 (m~-m2) ln~-~-12

+ ( ( J + i ~ ) m ~ + m ~ vZ2-v~ \ [m~ m~'] + v m~)ftM-~, " ~ , ]

v 2 - v , - (6+i~) m~+rn~ 2 2 5

× I ,

with the kinematical funct ionf(x , y) defined as

x l n x _ y__ylny. (37) f ( x , Y ) = l _ x

The same one-loop mechanism produces an electric dipole moment d~ and a magnetic moment #~. For solar neutrinos only Jd~+i/t~l is relevant [ 10]. Up to first order in 6, ~ this quantity is given by

edv sin 2~'

X (J+iE) ~m~+rn~ + ~-v2--rn

xJ(mL ML

( 2 + (6+ie) m~+m~ v~-v'f ~2"~ - - - - 2 - - + v 2 , , ,2)

XJ(m2, M~, M2) (38)

with the kinematical function

J(m2, M~,M~)

l l ( m2 m - M ~ ( 1 - m 2 / M ~ ) 2 l - ~ - - ~ + l n ~ - ~

(l-m2/M ) 2 o

(39)

In order to perform numerical estimates we assume that In (M2/MI) and (v~- v~)/v 2 are of order one. Consequently the neutrino mass is dominated by the first term in eq. (36) whereas for [d~+ig,{ the last term in eq. (38) is prevailing. Thus we obtain the very simple estimate

m~rn~J ( m 2 , M 2, M2) I (40)

where/tB = e/2me is the Bohr magneton. Taking li = e and using eq. (40) we get an upper bound of the order of Id~+iul < 10 -13 ~B for case (30a) (/2=/1). Thereby we have used Higgs masses close to the lower bound Mi> 19 GeV [ 11 ]. For case (30b) ( /2=z) the bound on Id ,+ i /~ l is somewhat smaller. A similar estimate of the neutrino moments in the Zee model is found in ref. [ 12]. It can easily be checked that I m~ I ~ 10 eV is compatible with the upper bound on the coupling constant d derived in ref. [3]. Though in our model the neutrino moments turn out to be too small to account for the solution of the solar neutrino problem as proposed by Voloshin, Vysotskii and Okun [ 13 ], we want to emphasize that I d~ + i u~ I is still much larger than in the standard model with right-handed neutrino singlets [ 14 ].

In this paper we have investigated the effect of a certain class of non-standard CP symmetries in the Zee model. In this way we have found a model with NFC and a ZKM lepton number which can both be induced by non-standard CP transformations. The corresponding Dirac neutrino gets a mass at the one- loop level. The remaining lepton generation with a massless neutrino obtains an independent lepton number. This is in contrast to the original Zee model where all neutrinos are of Majorana type. Further- more, in our model the mass splitting m , - m 2 between the charged leptons participating in the ZKM lepton number is connected to the spontaneous breakdown of the CP symmetry. This also shows that the present model is not identical with the Zee model with the usual CP invariance and additional lepton numbers L l - L 2 and L3 introduced by phase transformations, see eq. (22). In such a model m l - m z would be independent of spontaneous CP breaking. Finally, we have calculated the mass, the magnetic moment and the electric dipole moment of the Dirac neutrino. The mass can easily be of order 10 eV and there is an upper bound on the neutrino moments of order I 0-13/zB.

Wc thank G. Ecker and S.T. Petcov for reading the manuscript and valuable discussions.

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on spontaneous cp violation in the lepton sector

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