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Lawrence M. Krauss∗and Soo-Jong Rey**†

arXiv:hep-ph/9203212v2 17 Mar 1992Spontaneous CP Violationat the Electroweak ScaleLawrence M. Krauss∗and Soo-Jong Rey**†Institute for Theoretical PhysicsUniversity of California, Santa Barbara CA 93106**&Center for Theoretical Physics, Sloane LaboratoryYale University, New Haven CT 06511 USAFebruary 1992NSF-ITP-92-03YCTP-P9-92AbstractUtilizing results on the cosmology of anomalous discrete symme-tries we show that models of spontaneous CP violation can in prin-ciple avoid the domain wall problem first pointed out by Zel’dovich,Kobzarev and Okun. A small but nonzero θQCD explicitly breaks CPand can lift the degeneracy of the two CP conjugate vacua throughnonperturbative effects so that the domain walls become unstable, butsurvive to cosmologically interesting epochs.

We explore the viability∗also Department of Astronomy. Research supported in part by the NSF, DOE andTRNLC.

Bitnet: Krauss@Yalehep†Yale-Brookhaven SSC Fellow. Bitnet: Soo@Yalehep1

of spontaneous CP violation in the context of two Higgs models, andfind that the invisible axion solution of the strong CP problem cannotbe implemented without further extensions of the Higgs sector.2

Since the ground-breaking experiments on K0-decay in the 1960’s, ithas been recognized that the weak interaction violates CP invariance (andthus, assuming CPT, T as well) [1]. Nevertheless, in the intervening threedecades the mechanism responsible for (flavor non-diagonal) CP violationhas not yet been conclusively elucidated.Moreover, the recognition thatweak CP violation is communicated to the strong interaction via the QCDaxial anomaly has confused the issue further–especially with the lack of anyobservable electric dipole moments for the neutron and electron.A very simple possibility is that CP invariance is spontaneously brokenin conjunction with the breaking of other continuous global and/or gaugesymmetries.T.D.

Lee [2] was the first to point out that this mechanismis indeed possible through a complex vacuum expectation value (VEV) ofHiggs fields in a two Higgs doublet model of the SUL(2) ×UY (1) electroweakinteraction. His model gives rise to a flavor-changing neutral Higgs bosonexchange accompanied by the spontaneously broken CP invariances.

Thisleads to, for example, ∆S = 2 interactions at the tree level.In order to suppress flavor-changing neutral Higgs exchange interactions[3], Weinberg [4] proposed a class of multi-Higgs models. In this case, CPinvariance may be broken either spontaneously through complex Higgs VEVsor explicitly through complex-valued Higgs self-coupling constants (or both).Complex valued VEVs can also result naturally in theories without funda-mental Higgs particles.

For example, in technicolor models, a complex-valuedvacuum misalignment [5] of techniquark bilinear condensates [6] can occurat the electroweak scale.Of course, the simplest model of explicit CP breaking through the Higgscouplings resulting in the Cabbibo-Kobayashi-Maskawa (CKM) mass matrixis in good agreement with current CP violation phenomonology. Neverthe-less the idea that one might be forced beyond this minimal model has beenrevived with the recognition that the baryon number of the universe mightbe generated non-perturbatively at temperatures characteristic of the weaksymmetry breaking scale, i.e.

the electroweak baryogenesis scenario [7]. Inthis case, it has been claimed that new sources of CP violation, beyond thatembedded in the CKM mass matrix, will be required.In fact, the most serious argument against spontaneous CP violationprobably comes from cosmology.

In a seminal paper, Zel’dovich, Kozbarev3

and Okun [8] first pointed out that spontaneously breaking of a discretesymmetry such as CP in the early universe results in the formation of domainwalls during the phase transition associated with the symmetry breaking.Since these domain wall’s total mass is proportional to σR2(t), where σdenotes the domain wall mass per unit area and R(t) denotes the cosmicscale factor, their energy density scales as ≈1/R(t). In this case, the energydensity of domain walls can quickly come to dominate energy density inmatter and radiation, which scale as 1/R(t)3 and 1/R(t)4 respectively.One possible way out of a domain wall dominated universe is to assumethat the symmetry breaking scale is set higher than the scale at which infla-tion may occur so that domain walls are diluted away during an inflatinaryera.

In this case, feeding down CP violation to the low-energy physics worldrequires some clever model building [9]. On the other hand, if we preferthe scale of CP violation to be near the electroweak scale for the purposesof baryogenesis, or to explore possible new experimental signatures at cur-rent or future accelerators, this solution of the domain wall problem is notavailable.In this paper, we point out that because the discrete CP symmetry isanomalous due to the QCD axial anomaly, nonperturbative communicationbetween the fermion-Higgs sector and the QCD sector leads to a tiny butcosmologically significant splitting of the CP conjugate vacuum degeneracy,if we assume a small but nonzero θQCD(not ¯θ = θQCD + C2(R)ArgdetM)1.Therefore, as has been shown for such anomalous discrete symmetries [10],one can avoid the Zel’dovich et al.cosmological domain wall problem.

In thefollowing, we will illustrate this mechanism through a simplified model ofspontaneous CP breaking of two Higgs doublets. However, the same mecha-nism should apply to more realistic models.Let us start with Lee’s model [2] of two Higgs doublets that conserves fla-vor.

Neutral flavor conservation (NFC) is achieved, for example, by imposingGlashow-Weinberg’s Z2 discrete symmetryφ1 →φ1,φ2 →−φ2,Uor →Uor ,Dor →−Dor(1)1Note that most models of spontaneous CP breaking at high energy scales [9] werepreviously designed so that ¯θ = 0 at tree level.4

in which the quarks Uo, Do denote weak eigenstates. The most general, renor-malizable Higgs potential and Yukawa interactions respecting the Glashow-Weinberg Z2 symmetry readsLHiggs = −µ21|φ1|2 −µ22|φ2|2+ λ1|φ1|4 + λ2|φ2|4 + λ3|φ1|2|φ2|2+ λ4|φ†1φ2|2 + λ5[(φ†1φ2)2 + (φ†2φ1)2](2)andLY ukawa = (fij ¯QoLi ˜φ1UoRj + gij ¯QoLiφ2DoRj + h.c.).

(3)We assumed that the above terms are CP-invariant and thus all the couplingconstants are purely real-valued.As the electroweak symmetry is brokenby the vacuum expectation values, < φ1 ≯= 0, < φ2 ≯= 0, the Glashow-Weinberg discrete symmetry is also spontaneously broken. Therefore cosmo-logically dangerous Z2 domain walls arise at the phase transition.

However,this Z2 discrete symmetry is anomalous due to nonperturbative QCD effects[10]. In other words, the QCD instantons induce an effective local operator in-volving 2Nf quark flavors.

This operator is odd under the Glashow-WeinbergZ2 discrete symmetry, which is thus explicitly broken. Preskill et al.foundthat the cosmological domain wall problem can disappear due to this nonper-turbative violation of the Z2 symmetry.

Note that these arguments remainvalid irrespective of whether the CKM matrix is chosen to be real or not.Alternatively, one may resort to an explicit but small breaking of theGlashow-Weinberg Z2 discrete symmetry by adding the following termsδLY ukawa = ξ( ¯QoLif ′ij ˜φ2UoRj + ¯QoLig′ijφ1DoRj + h.c.)(4)andδLHiggs = ξ′(φ†1φ2 + φ†2φ1)(αφ†1φ1 + βφ†2φ2). (5)This will solve the cosmological domain wall problem associated with theGlashow-Weinberg’s discrete symmetry, without resorting to the QCD anom-aly.

In addition, CP remains a manifest symmetry of the Lagrangian. (Thismay also result in potentially unacceptable flavor changing neutral-Higgscurrents.In the context of this toy model, however, we will not concern5

ourselves about this problem. Phenomenologically viable two Higgs doubletmodel [11] or models with a richer Higgs structure can presumably avoid it.

)However, a new domain wall problem apparently results in this case.With the additional terms, CP is spontaneously broken. This is because,after the spontaneous electroweak symmetry breaking,φo1 = 1√2v1eiθ1,φo2 = 1√2v2eiθ2(6)in which the relative phase angle iscos(θ1 −θ2) ≡cos(κ) = −ξ′αv21 + βv224λ5v1v2(7)if we choose λ5 > 0.

As long as either one of ξ and ξ′ is nonzero, spontaneousweak-CP nonconservation arises. Since weak CP is spontaneously broken, anew kind of domain wall can result, which separates two CP conjugate worldsin the early Universe.

Such domain walls are cosmologically dangerous.The cure is, interestingly enough, connected with the QCD sector again,provided that a bare θQCD is nonzero and small (this ugly feature is merely arestatement of the ‘strong-CP’ problem). After the two Higgs fields get VEVsas in Eq.

(6), there is an induced flavor-diagonal CP violation, θQF D. The sizeof θQF D varies considerably with the Yukawa coupling constants. If ξ = 0, wefind θQF D(tree) ≈Ng(θ1−θ2) in which Ng denotes the number of generations.On the other hand, if ξf ′ ≈g and ξg′ ≈f, we find that θQF D(tree) ≈0.Nevertheless, there is generically a one-loop induced θQF D in the latter case,and is conservatively estimated to be θQF D(1 loop) ≈ξ′ GF16π2m2t ≈ξ′10−4for a top quark mass mt ≈100GeV .

With a reasonably small value of ξ′,θQF D(1 loop) can be as small as 10−9. Thus, in what follows, we assume thatboth θQCD and θQF D are typically of order 10−9 so that the ¯θ ≡θQCD +θQF Dremains small 10−9.At temperatures of the universe below the electroweak scale but wellabove ΛQCD, thermal suppression of QCD instanton effects renders the do-main wall practically stable.

The walls evolve and stretch out with expansionof the universe. As the temperature approaches ΛQCD, instanton effects turnon and yield a vacuum energy (in the zero temperature limit) of orderEvacuum ≈ΛQCDmumdms cos ¯θ.

(8)6

Under a CP transformation, θQF D →−θQF D. Therefore, the CP conjugatedegenerate vacua are now split in energy density by an amount∆Evacuum ≈ΛQCDmumdms sin θQCD sin θQF D.(9)Therefore, domain walls created at the electroweak phase transition beginto feel an energy difference between the two sides of the wall. This yields anon-zero pressure on the domain walls, and they begin to move as the falsevacuum decays to the true vacuum [12].We can use the arguments of Preskill et al.

[10] to estimate whether theabove energy difference is enough for the domain walls to decay away be-fore they start to dominate the energy density of universe. Reflections ofrelativistic particles offof the domain walls can result in an effective wallviscosity η ≈T 4 at temperature T, producing a dragging pressure p ≈T 4v.On the other hand, the curvature of the wall on a scale R(T) produces apressure ≈σR(T), which tends to straighten the wall.

Here σ is the wall ten-sion, which is roughly given by the mass per unit area of the wall. Thus,when the curvature induced pressure equals the viscous drag, irregularitieson a given scale to be smoothed out, as long as the time scale associatedwith motion on a scale R(T) is smaller than the Hubble time.

One findsthe critical straightening length scale Rs(T) ≈√σ/GNT 3. This is the minimallength on which wall segments will straighten out.

Since the wall energydensity is ρwall ≈σ/Rs(T), use of this value for Rs(T) results in the largestvalue of wall energy which has to be dissipated, and thus also in the mostconservative constraints on models. One findsρwallρrad≈qσGN1T≈10−8 1√λTEW.

(10)Thus, with a conservative value of the Higgs quartic coupling constant λ ≈10−4, the domain walls would start to dominate around T ≈300eV . At thenucleosynthesis scale, for example, the walls provide a negligible contributionto the total energy density of the universe.Let us see when the vacuum energy difference is large enough to drive thewalls from the true to the false vacuum regions.

The walls quickly move to7

the speed of light once the pressure provided by the vacuum energy differenceis larger than the viscosity ∆Evacuum ≥T 4. Assuming θQCD ≈θQF D ≈10−9and ∆Evacuum ≈ΛQCDmumdmsθQCDθQF D, this happens whenTd ≈10−5ΛQCD ≈1KeV.

(11)By the time the walls have reached the speed of light, the regions of falsevacuum are quickly driven away. In fact, this could easily occur when the wallvelocity is much slower.

At the time they start to move, the mean spacingbetween walls could be a small fraction of the horizon size. Even assumingrelativistic velocities are required, walls would be driven out before they startto dominate the energy density of Universe, as long as ¯θ ≥10−10−11.Note that the cosmological scales involved are quite interesting.

Thesedomain walls could remain in existence down to temperatures (i.e of O(KeV ))where present day galaxy sized regions first came inside the horizon. Theymight thus provide seeds for galaxy formation which might be relevant forlarge scale structure analyses.

One should also point out that while the do-main walls might contribute a small contribution to the total energy densityat the time they decay, their disappearance could result in energetic particleproduction. This could have interesting effects including possibly alter lightelement abundances produced during big bang nucleosynthesis (BBN) (i.e.see [13]).

Because there are various possibilities, including photo-dissociationof deuterium and helium, and also energetic baryon production which mightre-initiate some BBN reactions, one must investigate in detail the decay chainresulting from bubble wall collisions using explicit models for spontaneousCP breaking in order to make detailed predictions of what, if any effectsthere might be.The main unattractive feature in all of this is our assumption that ¯θis small, in the absence of any dynamical mechanism to make this so. Ofcourse, a natural solution to this strong CP problem is obtained by intro-ducing phenomenologically viable, invisible axions.

One might hope that theintroduction of a Peccei-Quinn mechanism might allow ¯θ to start out large(so that the domain walls associated with spontaneous CP violation at theelectroweak scale might be quickly removed) and that at a lower scale whenaxion mass effects turn on, ¯θ can relax to zero. We find that this cannot be8

easily achieved however.As pointed out by Preskill et al [10], incorporating a Peccei-Quinn sym-metry dynamically affects the cosmology of domain walls as discussed above.In the model introduced by Kim [14], there is an extra coupling involving anelectroweak singlet heavy quark and a singlet Higgs fieldLkim = g ¯QLQRΦ + c.c. (12)The weak isodoublet Higgs sector is largely unchanged, and spontaneous CPviolation can be accomodated as discussed above.

However, since the Peccei-Quinn symmetry: QR →eiθpqQR; Φ →e−iθpqΦ also suffers the same QCDanomaly as the discrete CP symmetry, one can find a linear combinationof these two anomalous symmetries to yield a new discrete, but nonanoma-lous symmetry. (The other remaining anomalous continuous symmetry isthe Peccei-Quinn symmetry, which can still solve the strong CP problem.

)This symmetry then suffers the standard domain wall problem when it isspontaneously broken by the Higgs VEVs at the weak scale.The other type of axion model due to Dine, Fischler and Srednicki andZhitniskii (DFSZ) [15] is also problematic. They introduced a singlet scalarfield Σ:LDF SZ = λP Qφ1φ†2Σ2 + h.c.(13)The Peccei-Quinn symmetry breaking < Σ >= vP Q gives rise to a term inthe low-energy effective Lagrangian λP Qv2P Qφ1φ†2 + h.c..

However, the Higgspotential and Yukawa coupling in Eqs. (2-5) do not respect the Peccei-Quinnsymmetry as long as any of λ5, ξ, ξ′ remain nonzero.

One would be requiredto set ξ = 0, and somehow fine tune λ5, ξ′ →0, while keeping their ratiofixed, in order for the terms leading to a possible spontaneous CP violationat the weak scale (see eq. (8)) not to also violently break the Peccei-Quinnsymmetry.

Thus, barring an apparently unnatural fine tuning, in the DFSZaxion model, it seems that the only viable option of CP violation is throughthe CKM mass matrix.Are there extensions of the DFSZ axion models that accomodate spon-taneous CP violation at the weak scale?Extensions involving either twoisodoublets and two isosinglets or three isodoublets and one isosinglet one9

might still lead to non-trivial CP violating Higgs VEV phases without intro-ducing extra non-anomalous discrete symmetries, or simultanously explicitlybreaking the PQ symmetry. Extensions of Higgs sector beyond the two Higgsmodels are also necessary to be phenomenologically realistic (e.g.

recall theproblem of flavor changing neutral currents). Such extended models mayhave several new interesting features and are currently under study.SJR acknowledges the hospitality of the Institute for Theoretical Physicsat Santa Barbara during the course of this work.

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