VOLUME 74, NUMBER 11 PHYS ICAL REVIEW LETTERS 13 MARcH 1995

Scheme for Radiative CI' Violation

Darwin Chang' " and Wai-Yee Keung2

'Physics Department, National Tsing-Hua University, Hsinchu, Taiwan2Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607-7059

High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-48}5Institute of Physics, Academia Sinica, Taipei, Taiwan

(Received 27 September 1994)

We present a simple model in which CP symmetry is spontaneously broken only after the radiativecorrections are taken into account. The model includes two Higgs-boson doublets and two right-handedsinglet neutrinos which induce the necessary non-Hermitian interaction. To evade the Georgi-Paistheorem, some fine-tuning of coupling constants is necessary. However, we show that such fine-tuningis natural in the technical sense, as it is protected by symmetry. Some phenomenological consequencesare also discussed.

PACS numbers: 11.30.Er, 11.30.Qc, 12.60.Fr, 14.60.St

More than 30 years after its experimental discovery, theorigin of CP violation still remains very much a mystery.In the widely accepted standard model and many othermodels, CP violation is a result of the complex parame-ters [1] allowed in the Lagrangian. For many physicists,such mundane explanation of the origin of the violation ofCP symmetry is not very satisfactory. In an effort to un-

derstand it at a deeper level, many different schemes havebeen conceived in the literature. A popular alternative isto require CP symmetry at the Lagrangian level and allowits nonconservation only in the vacuum. Such schemesare commonly termed spontaneous CP violation [2]. An-other even more ambitious attempt is to consider CP asan exact symmetry at the tree level but allow its noncon-servation only when the quantum effects are included. Torealize such a scheme within the perturbative framework,one naturally requires the CP violation to be spontaneousin origin also. Therefore, the model would have a Higgspotential in which its tree level ground states include aCP conserving one, but when the radiative corrections areincluded, a CP violating ground state is selected [3]. Inthat case, one can genuinely call the CP violation a quan-tum mechanical effect. It is the aim of this Letter to lookfor a realistic model of such type. Throughout this Letter,CP symmetry is assumed to be an exact symmetry of theLagrangian.

To implement this mechanism in any realistic model,there are two main obstacles. The first one is the Georgi-Pais theorem [4]. The theorem assumes that no fine-tuningof any kind is allowed. Under such an assumption, the firstconclusion one can make is that radiative CP violation ispossible only if the degeneracy of the ground states of thetree level Higgs potential is such that CP symmetry cannotbe asserted. That is, the CP violating ground states and theCP conserving ones are degenerate. Furthermore, Georgiand Pais also proved that radiative breaking can occur onlyif the tree level spectrum of the Higgs bosons contains amassless particle. Such a boson may eventually pick up

masses when the radiative effects are included. However,under the assumption of no fine-tuning such a boson isnecessarily light. Since the experimental limit on such alight boson is very strong, it seems very difficult to find arealistic model under such a scheme.

After the result of Georgi and Pais, there have been manyattempts to get around the constraint from the theorem.One can try to go beyond the perturbative framework [5]which is beyond our present scope. Alternatively, one canrelax the no-fine-tuning constraint and permit some fine-tuning as long as it is technically natural. (By technicallynatural, we mean, in this Letter, a set of parameterscan be assumed to be much smaller than the rest ofthe parameters as long as all the radiative corrections tothese small parameters naturally contain powers of theirsmall tree level values. ) However, even if the Georgi-Paistheorem is circumvented by technically natural fine-tuning,its physical origin can still present itself in the form of theexistence of a light Higgs boson in such a model.

An example of such a situation appears in the modelproposed by Maekawa [6]. In the minimal supersymmetricmodel, it is well known that spontaneous CP violationcannot happen at the tree level. However, Maekawashowed that, if some parameters in the Higgs potentialare much smaller than the gauge coupling, it is possible tohave spontaneous CP violation when the one-loop effectis taken into account. Following Maekawa, Pomarol [7]pointed out that the Higgs-boson spectrum of such a modelcontains a light boson whose mass lies in the range thathas already been ruled out by the data from the CERNe e collider LEP [8]. In general, the experimental boundon a pseudoscalar boson is not very strong because apseudoscalar boson does not couple linearly to the Z bosondirectly. However, in the minimal supersymmetric model,this bound becomes more severe as the pseudoscalar bosonmass can be related to the scalar boson mass.

Here we wish to present a simple Peccei-Quinn-typeextension [9] of the standard model in which the tree level

0031-9007/95/74(11)/1928(4)$06. 00 1995 The American Physical Society

VOLUME 74, NUMBER 11 PHYSICAL REVIEW LETTERS 13 MARcH 1995

vacuum is automatically CP symmetric, and the radiativecorrections induced by some of the Yukawa couplingscan produce a CP violating vacuum. The idea here isnot to champion a particular model, but to show that thefundamental mechanism underlying Ref. [6] has nothingto do with supersymmetry, and the problem [7] of a verylight Higgs boson facing the model in Ref. [6] is also notintrinsic to the mechanism itself.

As in Peccei-Quinn [9] or supersymmetric [10]models,we start with two-Higgs-boson doublets. In general, thefollowing non-Hermitian terms would appear in the Higgspotential:

~H1gg~ mlz@1 42 + ~5(41 42) + ~6 1t 1 $1 @1 @22 t 2 t t

+ ~7@1 @2@2rt'2 + H.c.t t (1)

Then we impose a Peccei-Quinn-type symmetry, Q1, toeliminate the non-Hermitian quartic terms of dimension 4.However, we shall allow the soft terms such as

2 t'—m12@1 @2 to break the Q1 symmetry. Beyond the treelevel, the Aq term as well as other non-Hermitian quarticHiggs couplings A6 and A7 will be induced as quantumcorrections.

At the tree level, since the only non-Hermitian couplingis m», the ground state is CP symmetric automatically asthe relative phase between (@1)and (pz) is zero. One maythink that, with the induction of non-Hermitian quarticterms from the Higgs-boson loops, it might be possibleto produce a CP violating ground state by fine-tuning.However, that is not the case because the induced quarticnon-Hermitian couplings will be proportional to m» andcannot be used to balance the tree level coupling, m», nomatter how much one tunes. Even worse, the sign of theleading contribution to A5 is negative. It was shown inRef. [6] that to get a CP violating ground state in the two-Higgs-doublet model it is necessary that the induced Aq

term is positive.To induce a positive A5 for our purpose, we need to

enlarge the particle content further. Here we choose toenlarge the leptonic sector by two additional right-handedneutrinos, N)g and N2g, in addition to the usual leptondoublet L. The spectrum of the model now looks like

L N1 Nz

the following three Majorana mass terms:

—Xsz1(dim-3) = P, 12N, CX2 + P11N, CN1

+ p, ~~N2 CN2 + H.c. (4)

It is important to note that a discrete symmetry,

Z2. N) ~ —N],

is respected by all terms in the Lagrangian except by theterms m» and p, ». Therefore it is natural to fine-tunethese two couplings m» and p, ]2 to be small. Also, oneshould keep in mind that the Majorana masses break thelepton number symmetry Qz softly. Note that P, 11, P,zz

mass terms are SU(2) X U(1) invariant. Therefore, theirvalues can in principle be much larger than the SU(2)breaking scale.

Before we get into the discussion of CP symmetrybreaking, it is also interesting to note that if one sets thecoupling m» to zero, one will get a divergent contributioninduced by the p, » term; however, the divergence is onlylogarithmic with a coefficient proportional to P, 12(P, 11 +P,zz) f1fz. In addition, since the couPlings A6 and A7 arealso forbidden by the Z2 discrete symmetry, their inducedvalues will be proportional to p, » or m» also. Therefore,by fine-tuning the parameters p, ]z and m]2 to be small onecan make all the couplings which are forbidden by the Z2

symmetry small (relative to the dimensional parametersP, 11 and P,zz). Near this limit, one can find a CP violatingground state.

To break CP symmetry spontaneously, the loop-induced Aq must have a positive sign. This can beachieved by the diagrams in Fig. 1:

2 2 2f»fzz P«PzzI

P11(6)2 ~

p 11 p'22 p 22

The positive sign of Aq can always be achieved when p, ]]and p, q2 are of opposite signs.

Y

Q1

Oz

1 1

2 00 0

0

l

(2)

where I' is the hypercharge and Qz is the lepton numbersymmetry, which is automatic as far as the dimension-4couplings are concern. The relevant Yukawa interactionsare

Pzz )(

~Y(N) f1LN1R 41 + fzLN2R 42 +

We assume the global Q1 symmetry on the hard(dimension-4) terms, but we allow the soft terms to breakthe symmetry. They contain, in addition to. the m» term,

FIG. 1. Feynman diagram for the induced vertex A5(@1pz)'via the fermion loop.

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VOLUME 74, NUMBER 11 PH YS ICAL REVIEW LETTERS 13 MARCH 1995

The minimization of the potential for the most generalcouplings has already been done in Ref. [6]. Parametriz-ing the vacuum expectation value (VEV) as (@~) =vie' /~2 and (P2) = v2/v 2, we obtain

2 2 22m)2 A6V] A7V2cos6 =

4A5v& v2(7)

N2RNIR

NI( 0 f)v)/2~2 f2vp/2~2 i

Ifi v i/~K& P 12I

( f~v2/&v & —, p iz pz2

where v~ is from the usual left-handed neutrino. Theneutrino spectrum is nothing but the usually seesawspectrum of one very light and two very heavy Majoranaparticles. This is especially true if one assumes thatthe singlet masses, )p, ii~, ~p, 22~, are much larger than theSU(2) breaking scale (while )p, iz(, (((p, ii(, )p, z2[, is fine-tuned to be small). Increasing the number of generationsby adding an index to v~ is going to simply increase thenumber of light Majorana particles.

Next, we deal with the problem of a potentially lightpseudoscalar boson A with mass mA = Q2Aq v sin6 in

2 2 1/2our model where v = v~ + v2. The value of A5 in

Eq. (6) can be naturally as large as 0.1. mz is easilyaround 30 GeV. The masses of the other scalar bosons areusually much larger. The potential limit on the mass of apseudoscalar Higgs boson comes from LEP experiments.However, in all the analyses [8], the pseudoscalar bosonsare assumed to be produced by the decay of a scalar bosonH. For the case when the scalar boson is very heavy(such as mH ) mz), no limit on m~ has been extracted yet.One may try to obtain a limit on the pseudoscalar bosonby considering the emission Z ~ Z*AA ~ l+l AA [11];however, the branching ratio is about 10 8, too small for

[The condition that the vacuum preserves U(1)EM is alsoanalyzed in Ref. [6].] First of all, since A6v& and A7vz aresimultaneously one-loop induced and Z2 breaking, theyare naturally small compared with m&2, which is a treelevel Z2 breaking term, and therefore negligible. To havea significant CP violating phase 6 we need m~2 to be ofthe same order as A5V~ v2. This requires the fine-tuning ofthe tree level coupling m]2. The fine-tuning is technicallynatural if we simultaneously fine-tune both m» and p, &2

because they are the only two tree level terms forbiddenby the Z2 symmetry. Therefore, one has arrived at amodel in which CP symmetry is spontaneously brokenby the radiative correction. (In contrast, it is easy to showthat the supersymmetric models such as the one proposedby Maekawa [6] are not technically natural. )

Since we have chosen to extend the lepton sector ofthe standard model, we shall next discuss the structureof the neutrino masses in the model ~ Consider the one-generation case. The Majorana mass matrix of the modelcan be written as

c

the present LEP data unless the ZZAA gauge vertex is verylarge for some peculiar reason which does not happen inthis model. A pseudoscalar Higgs boson lighter than a bquark can be ruled out by b sA [12].

Note that in the limit that the Higgs potential hasa custodial SU(2) X SU(2) symmetry, the pseudoscalarboson mass is the same as the charged Higgs-boson massat the tree level [13]. Of course, in our case, not onlythe Higgs potential contains a parameter which does notrespect the custodial symmetry. The CP violating groundstate we obtained also breaks the custodial symmetry.These breakings can contribute to the p parameter at theone-loop level; however, the resulting constraints [13] arenot significant numerically for the model considered here.

Finally, we shall make a short discussion of the CPphenomenology. The details of the CP phenomenologydepend on how the Higgs doublets are coupled to thequarks [14—16]. Since we have only touched upon theleptonic sector to produce radiative CP violation so far,there is some arbitrariness in deciding how the quarks arecoupled. Basically, these doublets can couple to quarksin two different ways. The first way is to couple one ofthe Higgs doublets to the up-type quarks uR and the otherone to the down-type quarks d&. This is the way chosenin the Peccei-Quinn mechanism [9]. The second way isto couple both types of quarks u&, d& to one and the samedoublet. We shall only discuss the first option here, eventhough the second option may also be interesting.

The leading mechanism of CP violation is through neu-tral Higgs-boson exchange. Since the tree level couplingsof the neutral Higgs bosons are Aavor conserving, theleading contribution to the CP violating e parameter inthe kaon system is through the two-loop diagrams [6,16].The Inechanism also tends to give a large contributionto the neutron electric dipole moment, d„. While it isgenerally believed that the neutral Higgs-boson exchangealone is not enough to account for all the known CP phe-nomenology [15], there are, however, some claims in theliterature [16] that, by properly adjusting parameters, itis possible to produce large enough e with small enoughd„ in some models of CP violation mediated by neutral

Higgs bosons. We shall not get deeply into this compli-cated and detailed phenomenological issue here becauseit is not really directly connected to the main issue wewish to illuminate. If it is indeed the case that some treelevel flavor changing neutral currents are needed to pro-duce large enough e, it can easily be accommodated inthis model by a small extension of the quark sector suchas adding a vectorial down quark [17] which appears in

E6-type grand unified theories. It is also known that theCP violating e' of the kaon decay and CP violating pa-rameters in hyperon decays are both negligible in this typeof model. Detailed analysis of this and various other pos-sibilities will be presented elsewhere. Finally, the strongCP issue in models with soft breaking of Peccei-Quinnsymmetry is discussed in Ref. [18].

VOLUME 74, NUMBER 11 PHYSICAL REVIEW LETTERS 13 MARcH 1995

Note that even if one does not impose CP symmetryon the Lagrangian, the Higgs sector alone is automati-cally CP conserving at the tree level. Therefore, evenin models with other sources of CP violation (such asthe Kobayashi-Maskawa mechanism [1]),quantum effectscan produce a new independent source of the CP viola-tion. Of course, in that case, one can no longer claim thatthe CP violation is a quantum mechanical effect.

Finally let us address the natural scale for the singlet.Taking the simplifying assumption that v

&

—v2 —v,f~ —f2 —f, and p, ~~

—p, 22—p„we can correlate the

pseudoscalar mass mq —f v/47r and the light neutrinomass m, —f2v2/p, in the relation p —4rrv(m~/tn, ).For m, —10 eV, one needs p, —10' GeV. In a grandunified theory, the singlet scale p, is presumably related tothe grand unified scale or an intermediate scale. Thereforehaving a high singlet scale is not a serious problem.To detach from the unification issue, the above problemof large p, can also be avoided by allowing the heavyleptons, N;R, only to couple to a heavier fourth generation.It is quite easy to implement such a proposal. Realisticmodels of such type will be provided elsewhere.

To conclude, we have shown that if one allows fine-tuning which is technically natural, it is relatively easyto construct models in which the tree level vacuum isCP invariant while the loop-corrected potential producesa CP nonconserving vacuum. The basic ingredient isto impose enough symmetry (the global Q& symmetryin our example) on the higher dimensional terms suchthat the tree level potential has only one soft, non-Hermitian, symmetry breaking term. Then the loop-induced higher order term can produce the desired CPnonconserving vacuum through fine-tuning. To make thefine-tuning technically natural, one then has to find asmaller symmetry (Zz in our case) which can forbid someof the soft terms while allowing the others. Since thesofter terms typically have smaller discrete symmetry thanthe hard terms, such symmetry is not too hard to findeither in general. The example we provided in this Letteris not only simpler than the supersymmetric models in theliterature, it is also more appealing because the necessaryfine-tuning in our case is technically natural. It alsodemonstrates that the scheme has sufficient flexibility tobe realistic.

We wish to thank the HEP theory group at ArgonneNational Laboratory for the hospitality and support while

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this work was conducted. We also wish to thank AlexPomarol, Chuan-Hong Chen, and Goran Senjanovic forvaluable discussions. This work was supported in part byResearch Corporation, in part by the U.S. Department ofEnergy, Division of High Energy Physics, under ContractsNo. DE-FG02-84ER40173 and No. W-31-109-ENG-38,and in part by National Science Council of Taiwan GrantNo. NSC 84-2112-M-007-042.

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