On the Anomalous Discrete Symmetry

arXiv:hep-ph/9209234v1 14 Sep 1992SFU-Preprint-92-7On the Anomalous Discrete SymmetryZheng HuangDepartment of Physics, Simon Fraser UniversityBurnaby, B.C., Canada V5A 1S6AbstractWe examine an interesting scenario to solve the domain wall problem recently sug-gested by Preskill, Trivedi, Wilczek and Wise. The effective potential is calculated inthe presence of the QCD axial anomaly.

It is shown that some discrete symmetriessuch as CP and Z2can be anomalous due to a so-called K-term induced by instantons.We point out that Z2domain-wall problem in the two-doublet standard model can beresolved by two types of solutions: the CP-conserving one and the CP-breaking one.In the first case, there exist two Z2-related local minima whose energy splitting is pro-vided by the instanton effect. In the second case, there is only one unique vacuum sothat the domain walls do not form at all.

The consequences of this new source of CPviolation are discussed and shown to be well within the experimental limits in weakinteractions.PACS numbers: 12.15.Cc, 11.30.Er, 11.30.Qc.August 24, 20211

Recently, Preskill, Trivedi, Wilczek and Wise (PTWW) [1] have reported an interestingscenario to solve the cosmological domain wall problem associated with spontaneously brokendiscrete symmetry. They have pointed out that because some discrete symmetry can beanomalous due to the QCD axial anomaly and instantons, a non-perturbative communicationbetween the Higgs sector and the QCD sector leads to a tiny but cosmologically significantsplitting of the vacuum degeneracy.

Incorporating PTWW’s idea, Krauss and Rey [2] haveshown that certain models of spontaneous CP violation can in principle avoid the domainwall problem provided that CP is slightly broken by θQCD in strong interactions. In thisletter, we examine the idea by computing the effective potential for Higgs bosons in thepresence of QCD chiral anomaly.

We show that the instanton dynamics for light quarksdoes break Z2symmetry of the two-doublet standard model. However, it may also lead to aspontaneous CP symmetry breaking.To illustrate how the anomalous discrete symmetry arises, let us first consider a simplestmodel with spontaneous CP violation.

The prototype of this model was first considered byT. D. Lee [3] where the Higgs field ϕ belongs to a real representation,L0 = 12(∂ϕ)2 −λ2(ϕ2 −η2)2 + ¯ψ(̸∂+ m −ifγ5ϕ)ψ.

(1)The minimum of the potential corresponds to ⟨ϕ⟩= ±η and CP symmetry is spontaneouslybroken. It was first pointed out by Kobsarev, Okun and Zeldovich (KOZ) that the degeneracyof CP conjugate vacua ⟨ϕ⟩= η and ⟨ϕ⟩= −η results in a serious domain wall problem incosmology [4].

However, the situation is quite different if the fermion field ψ suffers fromnon-abelian gauge interactions. In that case, (1) can be extended to include, for example,color interactions (ϕ is of course colorless)L = 12(∂ϕ)2 −λ2(ϕ2 −η2)2 + ¯ψ(̸D + m −ifγ5ϕ)ψ −14F 2 −iθF ˜F(2)where F ˜F =g232π2ǫµνρσF µνF ρσ.

Though CP is explicitly broken by the θ-term if θ ̸= 0, π,the domain wall problem persists at the tree level because the vacua ⟨ϕ⟩= ±η are still indegeneracy. However, we show that the degeneracy of the vacua will be lifted by taking intoaccount the chiral anomaly or the instanton effect.2

The effective action of the Higgs field is calculated asZ =ZD(ϕ)e−S0[ϕ]ZD(A, ¯ψ, ψ)e−S[ ¯ψ, ¯ψ;A;ϕ] =ZDϕe−Seff[ϕ](3)whereSeff[ϕ] = S0[ϕ] + ∆S[ϕ](4)and the quantum correction is given∆S[ϕ] = −lnZD(A; ¯ψ, ψ)e−S[ ¯ψ, ¯ψ;A;ϕ] ≡−ln ˜Z. (5)The calculation of ˜Z[ϕ] in the instanton field follows the standard semiclassical approximationmethod as illustrated in, e. g. , Ref.

[5]˜Z[ϕ] =XνZD[Acl]νe−S[Acl]det−1/2MAdetMψdetMgh(6)whereMA=−D2 −2FMgh=−D2(7)Mψ≠D + m −ifγ5ϕand ν stands for the winding number of the non-trivial topological gauge configuration. Ifthe effective potential is of concern, we can take ϕ in Mψ as a constant field.

The newphysics comes from the zero modes of the fermion determinant in the instanton field Acl.We factorize detMψ as followsdetMψ = det(0)Mψdet′Mψ(8)where “det(0)” denotes contributions from the subspace of zero modes of ̸D. According tothe index theorem [6], ̸D has a zero mode with chirality −1 (γ5 = −1) in a single instantonfield [7].

Thus we havedet(0)Mψ = m + ifϕ. (9)3

The prime in det′Mψ reminds us of excluding zero modes from the eigenvalue product.Since [̸D, γ5] ̸= 0, Mψ cannot be diagonalized in the basis of eigenvectors of ̸D. The non-vanishing eigenvalues of ̸D always appear in pair, i. e. if ̸Dϕn = λnϕn where λn ̸= 0, then̸Dγ5ϕn = −γ5 ̸Dϕn = −λnγ5ϕn, namely both λn and −λn are eigenvalues of ̸D.

In addition,γ5 takes ϕn to ϕ−n. Thereforedet′Mψ=Yλn>0detiλn + m−ifϕ−ifϕ−iλn + m=Yλn>0(λ2n + m2 + f 2ϕ2)=det′1/2(−̸D2 + m2 + f 2ϕ2),(10)i. e. det′Mψ is a function of ϕ2 which does not break the discrete symmetry.

It is to beemphasized that the above analysis does not depend on the detail of the instanton dynamics.It is the result of using the index theorem, which represents the general feature of the chiralanomaly in a gauge theory.Though we could proceed to analyze in general the effective potential based on Eqs. (9)and (10), we still would like to obtain the concrete form of Veff in the dilute gas approxi-mation (DGA) [8].

In the DGA,˜Z[ϕ] = det(−∂2 + m2 + f 2ϕ2) exp(˜Z+ + ˜Z−)(11)where˜Z+[ϕ]=V Keiθ(m + fϕ)˜Z−[ϕ]=V Ke−iθ(m −fϕ)(12)andK = 1.34CNcZ dρρ4 8π2g2(ρ)!2Nce−8π2g2(ρ). (13)ρ is the instanton density, CNc = N2c −12Nc , Nc is the number of colors.

In deriving (12), we haveassumed that m+ f⟨ϕ⟩is small compared to ΛQCD. Noticing that ln det(−∂2 + m2 + f 2ϕ2)contains terms which can be absorbed into the tree level lagrangian by redefining λ2 and η,we obtain the following effective potential (strictly speaking in the large Nc limit)Veff = λ2(ϕ2 −η2)2 + Keiθ(m + fϕ) + Ke−iθ(m −fϕ).

(14)4

Clearly, the last two terms (we shall call them the K-term) explicitly break CP symmetrywhen θ ̸= 0, for they are not invariant under TϕT −1 = −ϕ. The split in the energy densitybetween the CP conjugate vacua ⟨ϕ⟩= η and ⟨ϕ⟩= −η is given∆Evac = |Veff(η) −Veff(η)| = 4Kf sin θ|⟨ϕ⟩|.

(15)Therefore, domain walls created at the scale ⟨ϕ⟩will feel an energy difference between thetwo sides of the wall. The false vacuum at some space point will begin to decay towards thetrue vacuum.Another perhaps more interesting example to observe the anomalous discrete symmetryis to consider the two Higgs doublets model, which is the simplest allowed extension of thestandard model.

To achieve the natural neutral flavor conservation (NFC) at the tree level,we impose Glashow-Weinberg’s Z2discrete symmetry: φ1 couples with the charge 23 quarks(UR) and φ2 couples to the charge −13 quarks (DR), i. e. , for example,φ1 →φ1,φ2 →−φ2;UR →UR,DR →−DR. (16)The most general, renormalizable Higgs potential and Yukawa interactions which respect(16) readV0(φ1, φ2)=−m21φ†1φ1 −m22φ†2φ2 + a11(φ†1φ1)2 + a22(φ†2φ2)2(17)+a12(φ†1φ1)(φ†2φ2) + b12(φ†1φ2)(φ†2φ1) + λ[(φ†1φ2)2 + (φ†2φ1)2]andLY = ¯QLfUURφ1 + ¯QLfDDR ˜φ2 + h.c.(18)Here fU and fD are 3×3 Yukawa coupling matrices in flavor space, ˜φ2 = iσ2φ∗2.

The hermicityof V0 requires all coefficients in (17) are real. We shall examine the spontaneous CP violation(SCPV) in this model, thus we first choose fU and fD to be real and θQCD = 0 in the QCDsector.

When φ1 and φ2 acquire VEV’s, Z2symmetry (16) is spontaneously broken, whichposes dangers for cosmology. PTWW have argued that when the non-perturbative QCDeffect turns on, it breaks Z2symmetry and solves the domain wall problem.5

To see explicitly how PTWW’s idea works, we attempt to compute the K-term in theeffective potential following the same procedure as we did in the previous model. I will firstconsider one generation of light quarks consisting of u and d (mu, md ≪ΛQCD) to simplifythe problem.

The Higgs coupling to light quarks can be rewritten in a formLm = ( ¯uL¯dL ) HuRdR+ ( ¯uR¯dR ) H†uLdL(19)whereH ≡fdφ02∗fuφ+1−fdφ−2fuφ01. (20)Thus it is easy to identifyMψ ≠D + 12(H + H†) + 12(H −H†)γ5(21)where det Mψ runs over color, spinor as well as flavor indices,det(0)Mψ =det H† = fufdφ†1φ2for a single instantondet H = fufdφ†2φ1for a single anti-instanton(22)anddet′Mψ=det′1/2(−̸D2 + HH†)(23)=Yλn>0[λ4n + λ2n(f 2uφ†1φ1 + f 2dφ†2φ2) + f 2uf 2d(φ†1φ2)(φ†2φ1)].It is clear that det′Mψ can be absorbed into V0(φ1, φ2) in (17) but det(0)Mψ constitutes theso-called the K-term which breaks Z2symmetry.

The effective potential readsVeff(φ1, φ2) = V0(φ1, φ2) + Kfufd(φ†1φ2 + φ†2φ1)(24)whereK = (1.34)2CNcZ dρρ3 8π2g2(ρ)!2Ncexp −8π2g2(ρ)!. (25)K is of dimension 2.6

When λ < 0 (λ is the coefficient of [(φ†1φ2)2 + (φ†2φ1)2] in (17)), it can be readily shownthat the Z2-related (v1, v2) and (v1, −v2) (where v1 and v2 are real) are local minima ofVeff(φ1, φ2). However, they are not degenerate because of the K-term.

The difference in theenergy density between these two vacua (v1, v2) and (v1, −v2) is given∆Evac = |Veff(v1, v2) −Veff(v1, −v2)| ≃4Kfufdv1v2 = 4Kmumd. (26)K is the vacuum-to-vacuum amplitude in the instanton field.

It is also the amplitude ofthe axial U(1) symmetry breaking in QCD needed to solve the U(1) problem. It has beenestimated in [10] in connection with the U(1) particle massK ∼(m2η −m2π).

(27)Thus ∆Evac ≃10−4 ∼10−5GeV4, which is tiny but significant enough to solve the domainwall problem associated with Z2symmetry [1]. When λ > 0, none of (v1, v2) and (v1, −v2)are minima.

In fact, they are both local maxima of Veff. The true vacuum, denoted by(v1, v2eiα), which minimizes the effective potential acquires a non-trivial phase α (α ̸= 0, π).The domain wall problem associated with Z2is automatically resolved since (v1, −v2eiα) isno longer the minimum of the potential.However, what interests us is that the existence of the relative phase between ⟨ϕ1⟩and⟨ϕ2⟩breaks CP symmetry spontaneously.

To see this, we calculate the α-dependent termsin the effective potentialVeff(α) = 2λv21v22 cos 2α + 2Kfufdv1v2 cos α. (28)By minimizing Veff(α) with respect to α one obtainscos α = −Kmumd4λv21v22,(29)which is about 10−14 if v1 and v2 are taken to be the electroweak scale.

Therefore, Triggeredby strong interactions, CP is spontaneously broken at the electroweak scale in the two-doublet model. It is well known that Z2symmetry in the two-doublet model actually forbids7

the spontaneous CP violation. However, when Z2is explicitly broken by instanton effects,the SCPV is allowed but with a dynamically determined magnitude.Does this new source of CP violation lead to any observable effects in electroweak in-teractions?

Obviously, the phases of quark masses and Yukawa couplings originating fromSCPV can be rotated away by making appropriate hypercharge transformation. Thus theCP-breaking Cabbibo-Kobayashi-Maskawa (CKM) matrix does not arise in this model.

TheCP nonconservation is entirely due to neutral Higgs boson exchanges, i. e. through the mix-ing between scalar fields and pseudoscalar fields while the mixing probability is proportionalto sin α cos α which is about 10−14. All CP-violating processes are to be suppressed by thisfactor.

Its contribution to KL →2π can be neglected since this process involves charged fla-vor changing. The electric dipole moment of neutron (NEDM) will be receiving suppressionfactors, a 10−8 from Higgs propagators if Higgs bosons are of 100GeV and a 10−14 form themixings.

Thus the NEDM is estimated to be 10−34e·cm, which is not practically detectable.It is also not sufficient to generate the electroweak baryogenesis based on the weak phasetransition since the instanton effect is greatly suppressed at temperature characteristic of theweak scale [11]. Even though there are several ways of enhancing the CP violating effects by,for example, allowing a large ratio of v1 to v2 or having nearly degenerate masses for Higgsbosons, it would seem unnatural to yield any sizable observations.The evaluation of the relative phase between two vacuum expectation values can bereadily generalized to including any number of quark generations and explicit CP violationin a manner of KM mechanism without resorting to the instanton computations.

In general,the Yukawa coupling matrices fU and fD can be complex. The phases of their determinantscan be rotated away by redefining the right-handed quark fields while in the meantimechaning θQCD, the coefficient of the QCD θ-term.

We parametrize φ1 and φ2 in terms oftheir phase fields α1(x) and α2(x) asφ1 −→v1eiα1(x);φ2 −→v2eiα2(x)(30)and denote the relative phase field by α(x) ≡α1(x) −α2(x). The α1- and α2-dependence ofthe Yukawa couplings can be removed by making the local chiral rotations.

Because of the8

chiral anomaly, the θ-term becomesθQCD + nGα(x)F ˜F(31)where nG is the number of the quark generations. The effective potential for α(x) is calculated[12]Veff = −⟨⟨ν2⟩⟩QCD cosθQCD + nGα+ 2λv21v22 cos 2α(32)where the topological susceptibility ⟨⟨ν⟩⟩QCD is defined⟨⟨ν2⟩⟩QCD =Zd4x⟨TiF ˜F(x) iF ˜F(0)⟩.

(33)By minimizing (32) one obtainssin 2α =nG⟨⟨ν2⟩⟩QCD sin ¯θ4λv21v22(34)where ¯θ = θQCD + nGα.It is then clear from (34) that when θQCD ̸= 0, α = 0, π are not extremes of Veff(α), i. e.both (v1, v2) and (v1, −v2) are not local minima of the effective potential. The Z2domain-wallproblem is resolved by admitting a CP-violating solution (v1, v2eiα).

In this case, the weakCP violation is further suppressed by the requirement that sin ¯θ < 10−9 from the strong CPviolation. When θQCD = 0 and λ < 0, (34) gives the solution provided by PTWW, i. e.both (v1, v2) and (v1, −v2) are local minima whose energy splitting is caused by instantoneffects.

When θQCD = 0 and λ > 0, (34) allows a non-trivial CP-breaking solution α ̸= 0, π.In this case Z2domain walls do not form but domain walls associated with SCPV (α and−α) will begin to form.I would like to thank K.S. Viswanathan and D.D.

Wu for useful discussions.9

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