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The Measure of Strong CP Violation

arXiv:hep-ph/9209235v1 14 Sep 1992The Measure of Strong CP ViolationZheng HuangDepartment of Physics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6We investigate a controversial issue on the measure of CP violation in strong inter-actions. In the presence of nontrivial topological gauge configurations, the θ-term inQCD has a profound effect: it breaks the CP symmetry.

The CP-violating amplitudeis shown to be determined by the vacuum tunneling process, where the semiclassi-cal method makes most sense. We discuss a long-standing dispute on whether theinstanton dynamics satisfies or not the anomalous Ward identity (AWI).

The strongCP violation measure, when complying with the vacuum alignment, is proportionalto the topological susceptibility. We obtain an effective CP-violating lagrangian dif-ferent from that provided by Baluni.To solve the IR divergence problem of theinstanton computation, We present a “classically gauged” Georgi-Manohar modeland derive an effective potential which uniquely determines an explicit U(1)A sym-metry breaking sector.

The CP violation effects are analyzed in this model. It isshown that the strong CP problem and the U(1) problem are closely related.

Somepossible solutions to both problems are also discussed with new insights.PACS numbers: 11.30.Er, 12.40.Aa, 11.15.KcTypeset Using REVTEX1

I. INTRODUCTIONThe discovery of instantons [1] has been associated with some of the most interestingdevelopments in strong interaction theory. It has led to a resolution [2] of the long-standingU(1) problem [3], and also pointed to the existence in QCD [4] of vacuum tunneling and ofa vacuum angle θ, which combining with the phase of the determinant of the quark massmatrix, signals the CP violation in strong interactions.

The difficulty of understanding thevery different hierarchies of the strong CP violation and weak CP violation in the standardmodel has been targeted as the so-called strong CP problem (for a review, see Ref. [5]).The theoretical understanding of weak CP violation is well-established in the frameworkof Kobayashi-Maskawa mechanism [6] in spite of the challenge on the experiment measure-ment with higher precisions.

It has been shown [7] that the determinant of the commutatorof the up-type and down-type quark mass matrices [Mu, Md] ≡iC given bydet C = −2Jweak(mt −mc)(mc −mu)(mu −mt)(mb −ms)(ms −md)(md −mb)(1.1)whereJweak ≡sin2 θ1 sin θ2 sin θ3 cos θ1 cos θ2 cos θ3 sin δ(1.2)is the unique measure of the weak CP violation. All CP-violating effects in weak interactionmust be proportional to det C. Even though the CP-violating phase sin δ can be of order 1,the physical amplitude is naturally suppressed by the product of Carbibo mixing angles.However, the measure of CP violation in QCD, which we shall denote as Jstrong, is not soclear.

It has been long realized that θQCD and phases of quark masses are not independentparameters in QCD. In the presence of the chiral anomaly [8], they are related through thechiral transformations of quark fields.

Thus Jstrong must be proportional to a combination¯θ = θQCD + arg det M(1.3)which is invariant under chiral rotations. It is well-known that if one of quarks is massless, ¯θcan be of an arbitrary value since one can make arbitrary rotations on the chiral field.

This2

suggests that the ¯θ- dependence of Jstrong disappears in the chiral limit. Thus in the caseof L = 2 where L is the number of light quarks, Jstrong has a formJstrong = mumdK sin ¯θ(1.4)where we have written sin ¯θ instead of ¯θ to take care of the periodicity of ¯θ.

Is there anyother common factor that we can extract from strong CP effects? Or, is K in (1.4) only akinematical factor which varies with different physical processes.To answer the question, we need to know whether there is another condition under whichthe strong CP violation vanishes.

Recently, the reanalysis of strong CP effects has shed somelight on this issue. Several authors [9] have pointed out by studying an effective lagrangianthat the conventional approach to estimating the strong CP effects is erroneous in conceptthough numerically it is close to the correct one.

They believe that strong CP violationshould vanish if the chiral anomaly is absent. We regard their work as constructive andenlightening.

However, the connection of the effective theory with QCD is not apparentin their approaches. Indeed, if the chiral anomaly is absent in QCD, the phases of quarkmasses can be retated away without changing the θ-term.

But it is not clear why θQCDdoes not lead to CP violation in strong interactions. In addition, the presence of the chiralanomaly in a gauge theory may not directly related to CP violation.

One example is QED.It is well-known that QED is a CP-conserving theory even if it is chirally anomalous, and,in principle, could have a θ-term and a complex electron mass term.In this paper, however, we show that the measure of strong CP violation does acquire afactor referred to as the measure of the non-triviality of the non-abelian gauge vacuum. It issimply due to the fact that the θ-term is a total divergence whose integration over space-timeyields a surface term.

It can be dropped offunless there are non-trivial gauge configurationsat the boundary. K in (1.4) will be shown to be the vacuum tunneling amplitude betweendifferent vacua characterized by the winding numbersν =Zd4xF ˜F ≡g232π2Zd4xFµν ˜Fµν(1.5)3

where a semiclassical method makes most sense to deal with it. To probe the property ofthe K-factor, we proceed to consider a classically gauged linear σ-model.

A derivation of aU(1)A sector of the model can be made by taking into account the fermion zero modes inthe instanton fields. Contrary to the conventional result [10,12] where K has a singularityin quark masses such that Jstrong is a linear function of the quark mass, our model clearlyshows that K is to be explained as the mass difference between the U(1) particle and pions.Thus, Jstrong has a formJstrong = mumd(m2η −m2π) sin ¯θ.

(1.6)In the context of the effective model, the strong CP effects can be explicitly calculated andvarious solutions to the strong CP problem will be discussed with new insights.The paper is organized as follows. In sect.

2, we discuss a long-standing problem raisedby Crewther [10,11] on whether the instanton is or not consistent with the anomalous Wardidentity (AWI). We find that the AWI does not put any constraint on the topological sus-ceptibility ⟨⟨ν2⟩⟩in QCD.

The AWI is automatically satisfied by instanton dynamics if thesingularity in the chiral limit of some fermionic operator is taken care of. Sect.

3 dealswith a instanton computation of ⟨⟨ν2⟩⟩in the dilute gas approximation. The vacuum align-ment equations of the quark condensates are derived based on the path integral formalism.Upon making alignment among strong CP phases, we rederive an effective CP-violatinglagrangian.

In sect. 4 we present a classically gauged linear σ-model.

In the semiclassicalapproximation, the instanton fields are integrated out. An effective one-loop potential isobtained by integrating over fermions in the instanton background where the fermion zeromodes are essential to yield an explicit U(1)A symmetry breaking.

The strong CP effectsand the U(1) particle mass are calculated in the model. Sect.

5 devotes to discussions onvarious possible solutions to the strong CP problem. Sect.

6 reserves for conclusions.II. DOES INSTANTON SATISFY THE AWI?4

THE TOPOLOGICAL SUSCEPTIBILITY ⟨⟨ν2⟩⟩Let us leave our discussion on Jstrong aside for a moment and turn to a problem whichturns out to be a key to understand both strong CP violation and U(1) problem. It has beenlong pointed out that the instanton physics, in some ways, suffers from some difficulties.

Itis well-known that the integration over the instanton size is of infrared divergence. It isfurther argued by Witten [13] that the semiclassical method based on the instanton solutionof Yang-Mills equation is in conflict with the most successful idea of1Nc expansion in QCD.The reason is that instanton effects are of order e−1g2 , and for large Nc, g2 is of order e−Nc,which is smaller than any finite power of1Nc obtained by summing Feynman diagrams.

Theseproblems, as they stand now, indeed reflect various defects in the instanton calculation (wewill come back to these points in later sections).However, there was another type of objections initiated by Crewther [10] followed by oth-ers [11], which would be even more serious if they were correct. For many years Crewther hasemphasized that the breakdown of U(1)A symmetry by the chiral anomaly and the instantonis related to the breakdown of the SU(L) × SU(L) symmetry.

The relation is representedby the so-called anomalous Ward identity. He claimed that the instanton dynamics failedto satisfy the AWI and one would still expect the unwanted U(1)A goldstone boson.

Theyfurther showed that the topological susceptibility defined as⟨⟨ν2⟩⟩=Zd4x⟨T iF ˜F(x) iF ˜F(0)⟩(2.1)when satisfies the AWI must be equal to m⟨¯ψψ⟩(m is the quark mass, we have assumedthat all quarks are of equal masses). As we shall see in sect.

3, ⟨⟨ν2⟩⟩is to be identified asthe measure of strong CP violation. If Crewther were right, it would seem that the strongCP is of no direct relation with the topological vacuum structure.To see where the problem lies, we carefully follow a path integral derivation of the AWI.Consider a fermion bilinear operator ¯ψLψR with chirality +2 (sum over flavor indices isunderstood).

Its vacuum expectation value (VEV) is formally given5

⟨ψLψR⟩= 1V ⟨Zd4xψLψR(x)⟩= 1V1ZZD(A, ¯ψ, ψ)Zd4xψLψR(x)e−S[A, ¯ψ,ψ](2.2)where the QCD action in Euclidean space isS[A, ¯ψ, ψ] =Zd4x ¯ψ ̸Dψ + m ¯ψψ + 14F 2 −iθF ˜F(2.3)and Z is the normalization factor, V is the volume of space-time. Under an infinitesimalU(1)A transformationψR →eiα(x)ψR;ψL →e−iα(x)ψL(2.4)the measure D(A, ¯ψ, ψ) will change because of the chiral anomaly.

However, the integral(2.2) will not change since (2.4) is only a matter of changing integral variables. (2.2) thenbecomes⟨¯ψLψR⟩=1V ZZD(A, ¯ψ, ψ)Zd4xe2iα(x) ¯ψLψR(x) exp{−S[A, ¯ψ, ψ] +iα(x)Zd4x[∂µJ5µ−2m ¯ψγ5ψ −2L F ˜F]}(2.5)where the U(1)A current J5µ = ¯ψγµγ5ψ.

The independence of ⟨¯ψLψR⟩on α(x) implies itsvanishing of the first derivative which yields the AWIZd4x∂µ⟨T J5µ(x) ¯ψLψR(0)⟩= 2mZd4x⟨T ¯ψiγ5ψ(x) ¯ψLψR(0)⟩+2LZd4x⟨T iF ˜F(x) ¯ψLψR(0)⟩−2i⟨¯ψLψR⟩. (2.6)Crewther’s arguments go as follows.

If there is no U(1)A goldstone boson coupling to J5µ,the l.h.s. of Eq.

(2.6) vanishes. In the chiral limit, the first term of the r.h.s.

would vanishtoo. Thus one has when m →0LZd4x⟨T F ˜F(x) ¯ψLψR(0)⟩= ⟨¯ψLψR⟩.

(2.7)The instanton dynamics assumes that the integration over the gauge field is separated intoa sum over gauge configurations characterized by the integer winding number ν in (1.5), i.e.R [dA] = PνR [dA]ν and ⟨¯ψLψR⟩= PνR ⟨¯ψLψR⟩ν. Eq.

(2.7) would then imply6

(Lν −1)⟨¯ψLψR⟩ν = 0. (2.8)By assuming the spontaneous chiral symmetry breaking caused by ⟨¯ψLψR⟩̸= 0, (2.8) cannotbe satisfied if ν is an integer.

Moreover, by noting thatd⟨¯ψLψR⟩dθ= iZd4x⟨T F ˜F(x) ¯ψLψR(0)⟩(2.9)one obtains(−i ddθ −1)⟨¯ψLψR⟩= 0⇒⟨¯ψLψR⟩θ = ⟨¯ψLψR⟩θ=0ei θL(2.10)which is unacceptable because the θ-dependence of ⟨¯ψLψR⟩would have a wrong periodicity2πL. Along the same line, one could derive the AWI for operator ¯ψRψL and F ˜F and combinethem with (2.6) to obtain⟨⟨ν2⟩⟩= m2L2Zd4x⟨T ¯ψiγ5ψ(x) ¯ψiγ5ψ(0) + mL2⟨¯ψψ⟩.

(2.11)By inspecting the first term in the r.h.s of (2.11) is of order O(m2), one would conclude that⟨⟨ν2⟩⟩was a linear function of m, which, again, contradicts with the instanton computation.We argue, however, that all these inconsistencies arise from dropping the first term ofthe r.h.s.of (2.6) in the chiral limit or treating it as a higher order term. The U(1)A fermionoperator ¯ψiγ5ψ, when the fermion fields are integrated out first as they should be, mayobserve a1m singularity in certain gauge configurations.

To see this, we first calculate theVEV of ¯ψiγ5ψ in a fixed background field Aµ. Upon the fermion integration, one has⟨¯ψiγ5ψ⟩A = Triγ5̸D(A) + m = 1mT(m2)(2.12)whereT(m2) = Triγ5m2−̸D2 + m2 = Triγ5m2−D2 + 12gσµνFµν + m2.

(2.13)It is easy to check thatddm2T(m2) ≡0, i. e. T(m2) is independent of m2. Thus it can becalculated in the limit m2 →∞[20]7

limm2→∞T(m2) = −iLZd4x trγ5( 12σµνFµν)2Zd4p(2π)4m2(p2 + m2)3= iL F ˜F(2.14)and therefore⟨¯ψiγ5ψ⟩A = −iLF ˜Fm . (2.15)It observes a pole at m = 0.

It is clear that m⟨¯ψiγ5ψ⟩A may be finite in the limit m →0 ifF ˜F is nontrivial. Performing the fermion integration for the first term of r.h.s.

of (2.6), weobtainmZd4x⟨T ¯ψiγ5ψ(x) ¯ψLψR(0)⟩=Zd4x⟨T Tr imγ5̸D + m! (x) Tr 1 + γ52(̸D + m)!

(0)⟩−⟨Tr imγ5̸D + m1 + γ521̸D + m!⟩(2.16)= −LZd4x⟨T iF ˜F(x) Tr 1 + γ521̸D + m! (0)⟩−i⟨Tr 12(1 + γ5)m−̸D2 + m2!⟩.

(2.17)Identifying the second term in (2.17) with ⟨¯ψLψR⟩, we find that the r. h. s. of (2.6) vanishesidentically for any m.This is not surprising since if we had considered a global U(1)Atransformation instead of a local one in (2.4) at the beginning, we would have come up withthe same conclusion immediately. Similarly, (2.11) is an identity to be satisfied (trivially)by any dynamics which respects the basic rule of the fermion quantization (and of coursethe anomaly relation.

If there were no anomaly, the second term of r.h.s. of (2.6) wouldbe absent.The cancellation would be incomplete indicating the existence of a masslessexcitation coupling with J5µ.

Thus the chiral anomaly is essential to solve the U(1) problem. ).There is a delicate problem about taking the chiral limit.

One may ask what if the quarkmass term is simply absent in the lagrangian at the first place. Crewther’s problem seemsto come back if the first term of the r. h. s. of (2.6) is not present.

Actually this is where thepuzzle comes about. In this case, however, a nonvanishing value of the quark condensateis not well-defined.

It relates to a general feature of the spontaneous symmetry breakingmechanism.For example, in the φ4-theory with spontaneous breaking of the reflectionsymmetry (φ →−φ), the VEV of φ is calculated8

⟨φ⟩= 1ZZdφ φ e−Rd4x(∂µφ)2+ λ4(φ2−v2)2. (2.18)Since the action is perfectly reflection-symmetric and φ is an odd operator under reflection,we have ⟨φ⟩≡0.

Mathematically this is true because of the equal weight of degeneratevacua. But what is of physical interest is a situation where one of the degenerate vacua ischosen as the ground state.

The way to do it is to introduce a source termR d4xJφ into theaction which breaks the symmetry explicitly. The degeneracy of the vacua in the absenceof the source implies that ⟨φ⟩J is a multi-valued function of J at J = 0.

The VEV’s of φcrucially depends on the way that J tends to zero. In particular, ⟨φ⟩J→0+ = −⟨φ⟩J→0−̸= 0.The same procedure should follow for the spontaneous chiral symmetry breaking inQCD.

In order to define the quark condensate ⟨¯ψLψR⟩, one ought to add the source termR d4xJ ¯ψlψR(x) to the action. Then a U(1)A transformation changes the source term as wellZd4xJ ¯ψLψR →Zd4xJe2iα ¯ψLψR.

(2.19)We also need to take this change into account because ⟨¯ψLψR⟩defined by the way thatJ →0 would be different from the one defined by Je2iα →0. By differentiating ⟨¯ψLψR⟩with respect to α we obtain a equation exactly the same as (2.6) except that m is replacedby J.

For the same reason as we have discussed, the r. h. s. of the equation is identicallyzero for any value J (even in the limit J →0). There is no U(1)A goldstone boson, and, ingeneral, (2.7), (2.8 and (2.10) do not hold.We have shown that the AWI for the isosinglet current J5µ is trivially satisfied by QCDdynamics including the axial anomaly.

(2.11) is an identity satisfied by any dynamics if thesingularity of the singlet operator ¯ψiγ5ψ in the zero mass limit is appropriately handled. Itdoes not put any constraint on how the topological susceptibility ⟨⟨ν2⟩⟩should behave asa function of the quark mass.

Thus, it does not, from the context of the field theory, ruleout the instanton computation. However, this should not be confused with the case of theAWI’s for non-singlet currents where the assumption on the lowest lying resonances haveto be made.

For a non-singlet axial current Jaµ = ¯ψγµγ5λa2 ψ (λa’s are generators of SU(L),a = 1, · · · , L2 −1), the corresponding AWI reads9

m2Zd4x⟨T ¯ψiγ5λa2 ψ(x) ¯ψiγ5λb2 ψ(0)⟩−δabmL ⟨¯ψψ⟩= 0. (2.20)It can be readily checked by integrating the fermion fields that (2.20) is satisfied in QCD.Unlike the singlet current in (2.12)⟨¯ψiγ5λa2 ψ⟩A = Trλa2iγ5̸D + m = 0(2.21)because λa’s are traceless.

Assuming that pions are lowest lying resonances which dominant,one obtainsm2Zd4x⟨T ¯ψiγ5λa2 ψ(x) ¯ψiγ5λa2 ψ(0)⟩res. = F 2πm2πδab(2.22)leading to F 2πm2π = −1Lm⟨¯ψψ⟩.

Can we do the same analysis for the singlet operatorm2Zd4x⟨T ¯ψiγ5ψ(x) ¯ψiγ5ψ(0)⟩res. =?

(2.23)such that we may get a phenomenological value for ⟨⟨ν2⟩⟩from (2.11) without resorting toinstanton computations? This turns out to be of some difficulties.

For the axial singletoperator, we cannot generally assume the pion dominance. In fact, m ¯ψiγ5ψ does not coupleto pions because λa’s commute with identity [12].

In addition, ¯ψiγ5ψ has pole behaviorat m = 0 whose residue is F ˜F. It may couple to a gauge ghost [14] as well as glue ballsand U(1)A particle.

It may also exhibit a non-zero subtraction constant in the spectraldispersion representation [15], which by itself is not surprising in the presence of anomaly.All these factors may further fall into overlap, causing double countings. These have madean estimation on (2.23) extremely difficult if not impossible.In summary, the AWI and the low energy phenomenology may not put a constraint onthe topological susceptibility.

Therefore, it leaves us a task of calculating ⟨⟨ν2⟩⟩and themeasure of strong CP violation from instanton dynamics. To avoid the infrared divergence,we further relate ⟨⟨ν2⟩⟩to the U(1)A particle mass in an effective theory.10

III. THE EFFECTIVE CP VIOLATING LAGRANGIAN IN QCDIn Sect.

2 we have shown that the axial singlet operator ¯ψiγ5ψ is related to F ˜F in a fixedgauge background. When the gauge fields are integrated out, (2.15) becomes a relation onVEV’s.

It can be easily proven that such a relation is true for each flavor. In general, whenthe quark mass is complex, one derives−i(mieiϕi⟨¯ψiLψiR⟩−mie−iϕi⟨¯ψiRψiL⟩)= −i(mieiϕi⟨Tr121 + γ5̸D + mieiϕiγ5 ⟩−mie−iϕi⟨Tr121 −γ5̸D + mieiϕiγ5 ⟩= ⟨iF ˜F⟩(3.1)where ϕi is the phase of the ith quark mass (i = 1, · · ·, L), no sum over i is understood in(3.1).

Now define⟨¯ψiLψiR⟩≡−Ci2 eiβi;⟨¯ψiRψiL⟩≡−Ci2 e−iβi(3.2)or⟨¯ψiψi⟩≡−Ci cos βi;⟨¯ψiiγ5ψi⟩≡Ci sin βi(3.3)where Ci > 0 and βi is the phase of the ith quark condensate. Eq.

(3.1) yields⟨iF ˜F⟩= −miCi sin(ϕi + βi). (3.4)(i = 1, 2, · · ·, L)which is to be referred to as the vacuum alignment equation (VAE) [17].

It can also bederived directly by taking vacuum expectation values on both sides of the anomaly relation[21]. Eq.

(3.4) means that if the first moment of the topological charge is non-zero in thepresence of instanton, the quark condensate develops a phase βi different from −ϕi. If thephase of the fermion mass ϕi is zero as it can always be made so by making a chiral rotation,the fermion condensate has a non-trivial phase βi ̸= 0 i. e. develops an imaginary part whichis determined by the topological structure of the theory.

This of course would not happen11

in a theory like QED where only the trivial topological configuration exists. We shall seethat it is the combination ϕi + βi’s that determine the CP violating amplitude in stronginteractions.⟨F ˜F⟩can be calculated from instanton dynamics in the dilute gas approximation (DGA)[16].

The vacuum to vacuum amplitude in the presence of the θ-term is givenZ(¯θ) =∞Xν=0,±1,···ZD(A, ¯ψ, ψ)ei¯θνe−Rd4xPi¯ψi(̸D+mi)ψi+ 14F 2(3.5)where we have not explicitly included the gauge fixing and the ghost terms. Inclusion ofthem must be understood when the practical computation is performed.

The phase of thequark masses have been rotated away and ¯θ = θQCD + Pi ϕi. In the DGA,Z(¯θ) =∞Xn+=0∞Xn−=01n+1n−(Z+)n+(Z−)n−= eZ++Z−(3.6)where Z+ (Z−) is the one single instanton (anti-instanton) amplitudeZ+ = ei¯θZd4zdρρ5 CNc( 8π2g2(ρ))2Nce−8π2g2(ρ)d(Mρ)Z−= Z∗+(3.7)withCNc =4.6 exp(−1.68Nc)π2(Nc −1)!

(Nc −2)!.The factor d(Mρ) in (3.7) is connected with the so-called fermion determinant, which in-troduces important physics. It was first discovered by ’t Hooft [18] that there exists a zeromode of the operator ̸D in the instanton field.

Thus we expect d(Mρ) ∝det M (M is thequarks mass matrix). For small quark masses, d(Mρ) is equal to [18,19]d(Mρ) =LYi=1f(miρ)(3.8)f(x) = 1.34x(1 + x2 ln x + · · ·),x ≪1.Combining (3.8) and (3.7) with (3.6) one obtainsZ(¯θ) = exp[2V cos ¯θm1m2 · · · mLK(L)](3.9)12

where K(L) is of dimension 4 −LK(L) ∼= (1.34)LZdρρ5−LCNc( 8π2g2(ρ))2Nce−8π2g2(ρ). (3.10)The first moment ⟨iF ˜F⟩is calculated by taking an average of the topological charge over4-space⟨iF ˜F⟩= 1V ⟨Zd4xiF ˜F⟩= 1Vdd¯θ ln Z(¯θ)= −2mumd · · · mLK(L) sin ¯θ(3.11)and the topological susceptibility is equal to⟨⟨ν2⟩⟩= 1Vd2d¯θ2 ln Z(¯θ) = −2mumd · · · mLK(L) cos ¯θ.

(3.12)Clearly enough, when ¯θ is small we have⟨iF ˜F⟩= ⟨⟨ν2⟩⟩¯θ. (3.13)The vacuum alignment in QCD can be readily made through the VAE (3.4).

By definingthe quark field, one can change the phase of the quark mass ϕi and phase of the quarkcondensate βi. However, ϕi + βi’s will not change under the redefinition.

They are onlyfunctions of ¯θ as shown in (3.4). One may choose βi = 0 (i = 1, · · ·, L) such that thevacuum is CP-conserving⟨¯ψiiγ5ψi⟩= 0.

(i = 1, 2, · · ·, L)(3.14)Then the phase of the quark masses are no longer arbitrary. They are uniquely determinedby the vacuum alignment equation (3.4),ϕi = −⟨⟨ν2⟩⟩miC¯θ(i = 1, 2, · · ·, L)(3.15)θQCD = ¯θ −Xiϕi = 1 −Xi⟨⟨ν2⟩⟩miC!¯θwhere we have assumed ϕi’s are small and Ci’s are all equal to C. To be aligned with thevacuum, the strong CP phase ¯θ must be distributed among the θ-term and the quark mass13

terms according to their determined weights. The effective CP-violating part of the QCDlagrangian readsLβi=0CP= iθQCDF ˜F −2C mumd · · · mLK(L)¯θ ¯ψiγ5ψ.

(3.16)with θQCD given in (3.15). (3.16) is different from that obtained by Baluni [23], which, asappearing in most literatures, lacks the topological factor K(L) and fails to observe thetopological feature of the strong CP violation.It is worth emphasizing that the effective CP-violating interactions in (3.16) are only validin the CP-conserving vacuum where βi’s are zero.

One can alternatively choose a certainpattern of the phase distribution and ask in what direction the vacuum is to align with it.In general, the vacuum angles are not zero and should be determined by the VAE (3.4). Forexample, we can choose ϕi = 0 (i = 1, · · ·, L) such that Lβi=0CP= i¯θF ˜F.

In this case, thevacuum condensates are complex βi = −⟨⟨ν2⟩⟩miC ¯θ. A physical CP-violating amplitude is fromboth CP-violating part of the lagrangian and CP-violating part of the quark condensate.

Aproof of the equivalence of different chiral frames on strong CP effects is given in Ref. [22]where it is shown that the vacuum alignment equation (3.4) plays an essential role.Does the left-over θ-term in the effective lagrangians play any role in computing the strongCP effects?

So far there have been only two CP violating processes available: η →2π andthe electric dipole moment (EDM) of neutron. The latter process depends on a computationon the effective CP-odd π-N coupling [24].

Both of them would involve in an evaluation ofthe commutator [Qa5, F ˜F] if the θ-term were to contributeDπaπb θQCDF ˜F |η⟩= −iθQCDFπDπb [Qa5, F ˜F] |η⟩;⟨πaN| θQCDF ˜F |N′⟩= −iθQCDFπ⟨N| [Qa5, F ˜F] |N′⟩(3.17)where we have used the soft-pion theorem. It is obvious that [Qa5, F ˜F] = 0 since Qa5 is anon-singlet charge and thus the canonical commutation relation applies.

It is at least safeto argue that the θ-term in the effective lagrangian can be ignored. What really matters isthe correlating feature of φi’s and βi’s given by (3.4).14

The above statement can be justified in the following example. For simplicity, let usassume mu = md = · · · = mL = m and L = 3 where pions and η are all light pseudoscalarsand the soft-pion theorem applies.

The amplitude of η →2π is readily calculated when βi’sare zeroA(η →2π) =Dπ0π0 Lβi=0CP |η⟩= ¯θ−iFπ3 D[Q35, [Q35, [Q85, ¯ψiγ5ψ]]]E= 4√31F 3πmumdmsK ¯θ(3.18)In deriving (3.18), we have dropped offF ˜F term. In a chiral frame where φi’s are zero,we still drop offthe θ-term.

But the CP-conserving part of the lagrangian will contributebecause the vacuum condensates are Cp violatingA(η →2π) = −Dπ0π0 m ¯ψψ |η⟩= −m−iFπ3 D[Q35, [Q35, [Q85, ¯ψψ]]]E= −2√31F 3πmC sin β = 4√31F 3πmumdmsK ¯θ(3.19)where βi = −⟨⟨ν2⟩⟩miC ¯θ. Both (3.12) and (3.19) yield the same result.We conclude that the measure of strong CP violation is given by the topological suscep-tibilityJstrong = −12⟨⟨ν2⟩⟩¯θ = m1m2 · · · mLK(L)¯θ(3.20)However, K(L) is still an unknown factor because the integral in (3.10) is simply divergent.This is the shortcoming of all instanton computations if one use the dilute gas approximation[25].

(3.20) can only make sense if one introduces a cutoff¯ρ at large instanton density. thisbrings in an ambiguity of choosing ¯ρ.

Fortunately, as we shall show below, such an ambiguitycan be removed by considering an effective model where K(L) can be naturally related tothe mass of the U(1)A particle.IV. THE EFFECTIVE CHIRAL MODEL15

A. The model and the Instanton Induced Quantum CorrectionsWe consider an effective chiral theory where meson degrees of freedom are explicitlyintroduced.

The virtue of the model is that it reflects all flavor symmetries in strong in-teractions as described by QCD. Since the mesons as independent field excitations coupleto fermions through Yukawa couplings, there is no need to saturate correlation functions ofvarious currents in QCD with unclear assumptions on the lowest-lying resonances.

Unlikea conventional effective theory [26] in which the nucleons are involved, the model that wewill be discussing contains quarks, gluons and mesons. It is a linear version of the gaugedsigma model suggested by Georgi and Manohar [27], which describes strong interactions inthe intermediate energy region between the scale of the chiral symmetry breaking and thescale of the quark confinement.The model readsL = −¯ψ ̸Dψ −14F 2 + iθF ˜F −f ¯ψLφψR −f ¯ψRφ†ψL −Tr∂µφ∂µφ† −V0(φφ†) −Vm(φ, φ†)(4.1)where φ is a complex L×L matrix, V0(φφ†) is the most general form of a potential invariantunder U(L) × U(L) (renormalizable)V0(φφ†) = −µ2Trφφ† + 12(λ1 −λ2)(Trφφ†)2 + λ2Tr(φφ†)2(4.2)andVm(φ, φ†) = −14meiχTrφ −14me−iχTrφ†.

(4.3)(4.1) needs some explanations. Under U(L)L×U(L)R, the quark fields as well as the complexmeson field transforms asψL →ULψL,ψR →URψR;φ →ULφU†R,φ† →URφ†U†L.

(4.4)16

In the absence of Vm, L is invariant classically under (4.4) but broken down to SU(L)L ×SU(L)R×U(1)V by the chiral anomaly. Vm, replacing the quark mass (m now is of dimension3), serves as an explicit symmetry breaking and must be treated as a perturbation.

f is theYukawa coupling, chosen to be real by redefining φ. Under U(1)A transformationψL →eiωψL,ψR →e−iωψR;φ →e2iωφ,φ† →e−2iωφ†.

(4.5)the θ-term and Vm change as θ →θ −2Lω, χ →χ + 2ω. But ¯θ = θ + Lχ keeps unchanged.Except the meson sector, the gauge interaction in (4.1) looks identical to QCD.

One maywonder if we are doubly counting the degrees of freedom. This is explained in [27] thatthese quarks and gluons are not the same as in QCD.

In particular, quarks are supposed toacquire constituent masses about 360MeV , which is huge compared to the current mass inQCD. The gauge coupling gs between quarks and gluons in the effective theory is found tobeαs ∼= 0.28(4.6)much less than its QCD counterpart.

This may explain why nonrelativistic quark modelworks since the quarks inside a proton could be treated as weakly interacting objects.However, the drawback of the model is that it has a very serious U(1) problem. Indeed, ifone calculates the physical spectrum from V0 + Vm, one finds L2 would-be goldstone modes.In addition, the nontrivial topological structure of the theory has been totally overlooked.The classical excitations such as instantons have not been accounted for in the model, which,according to the original idea of ’t Hooft [2], are crucial to solving the U(1) problem.We therefore consider the quantum correction to the lagrangian (4.1) in the presenceof non-trivial classical gauge fields known as instantons.

We argue that the effective gaugecoupling αs in (4.6) is obtained only if those classical extrema to the action have beeneffectively summed up by the semiclassical method.We find that the 1-loop quantumfluctuations around instantons lead to a dramatic change on the U(1)A sector of the model.17

The U(1) particle acquires an extra mass from the vacuum tunneling effects, which, in turn,results in the so-called strong CP problem.The effective action of the meson field is calculated asZ =ZD(φ, φ†)e−S0[φ,φ†]ZD(A, ¯ψ, ψ)e−S[ ¯ψ,ψ;A;φ,φ†]=ZD(φ, φ†)e−Seff [φ,φ†](4.7)whereSeff[φ, φ†] = S0[φ, φ†] + ∆S[φ, φ†](4.8)and the quantum correction is given∆S[φ, φ†] = −lnZD(A, ¯ψ, ψ)e−S[ ¯ψ,ψ;A;φ,φ†] ≡−ln ˜Z[φ, φ†]. (4.9)The calculation of ˜Z[φ, φ†] in the instanton background follows the standard derivation ofthe vacuum-to-vacuum amplitude as in [18]˜Z[φ, φ†] =XνZDAcleiθν−S[Acl](DetMA)−1/2DetMψDetMgh(4.10)whereMA = −D2 −2FMgh = −D2(4.11)Mψ = ̸D + f2(φ + φ†) + f2 (φ −φ†)γ5.If only the effective potential is of concern, φ and φ† in Mψ are to be taken as constantfields.

The fermion determinant, as usual, needs special treatment:DetMψ = Det(0)MψDet′Mψ. (4.12)“Det(0)” denotes contributions from the subspace of zero modes of ̸D.

In a single instantonfield, ̸D has a zero mode with chirality −1 (γ5 = −1) [20]. Thus we haveDet(0)Mψ = det"f2(φ + φ†) + f2(φ −φ†)(−1)#= det(fφ†)18

where “det” only acts upon flavor indices. The prime in Det′Mψ reminds us to excludezero modes from the eigenvalue product.

Since [̸D, γ5] ̸= 0, Mψ cannot be diagonalized inthe basis of eigenvectors of ̸D. The nonvanishing eigenvalues always appear in pair, i. e. if̸Dϕn = λnϕn where λn ̸= 0, then ̸Dγ5ϕn = −γ5 ̸Dϕn = −λnγ5ϕn, namely both λn and −λnare eigenvalues of ̸D.

In addition, γ5 takes ϕn to ϕ−n. ThereforeDet′Mψ = detYλn>0iλn + f2(φ + φ†)f2(φ −φ†)f2(φ −φ†)−iλn + f2(φ + φ†)= detYλn>0(λ2n + f 2φφ†) = Det′1/2(−̸D2 + f 2φφ†).

(4.13)Now we are ready to make the DGA. We need to further assume a weak-field approximationof φ and φ†.

This can be justified by imagining that φ and φ† fluctuate about the VEV, whichis about 300MeV . The large fluctuations are exponentially suppressed by exp(−λ1|φ|4).

Inthe DGA˜Z[φ, φ†] = Det1/2(−∂2 + f 2φφ†) exp( ˜Z+ + ˜Z−)(4.14)where˜Z+[φ, φ†] = eiθ det(fφ†)Zdzdρρ5 CNc 8π2g2(ρ)!2Nce−8π2g2(ρ)·deth1.34ρ1 + f 2φφ† ln f 2φφ† + · · ·i∼= V K(L)eiθ det(fφ†)(4.15)˜Z−[φ, φ†] = ˜Z†+[φ, φ†]and K(L) is given in (3.10).Combining (4.14) with (4.9), and noticing that ln Det(−∂2+f 2φφ†) contains terms whichcan be absorbed into the tree-level lagrangian by redefinition of bare parameters, we obtainthe following effective lagrangianLeff = −¯ψ ̸Dsψ −14F 2s −(f ¯ψLφψR + h.c.) −Tr(∂µφ∂µφ†) −V0(φφ†) −Vm(φ, φ†) −Vk(φ, φ†)(4.16)19

whereVk(φ, φ†) = −K(L)f Leiθ det φ† −K(L)f Le−iθ det φ(4.17)Several remarks on (4.16) are in order. The presence of Vk in (4.16) is the direct result offermion zero modes in the instanton field.

It is invariant under SU(L)L × SU(L)R × U(1)Vbut not invariant under U(1)A. Under U(1)A rotation (4.5), eiθ det φ →ei(θ−2ωL) det φ. ThusVk takes over the role of the θ-term and the anomaly.

Again, ¯θ = θ + χL remains invariant.The prototype of Vk was suggested long time ago by several authors [28] and re-discussedby t’ Hooft [29]. It is different from a model originally proposed by Di Vecchia [32] andrecently analyzed in Ref.

[9], although physical contents of both models may be similar.The gauge interactions between quarks and gluons are still present in (4.16) as required inthe nonrelativistic quark model. However, they differs from QCD in that the gauge couplinggs has a smaller value, and the most importantly, the gauge field As now possesses a trivialtopology at infinity.

The gauge interaction sector in (4.16) is very analogy to QED: thefermion chiral anomaly still exists, but any θ-termR d4xθFs ˜Fs in the action would be simplya vanishing surface term and can be dropped off.B. The U(1) Particle Mass and Strong CP ViolationWe would like to discuss the physical spectrum of the model (4.16) (this part has beenworked out in Ref.

[29]) and show how the strong CP effects can be calculated effectively.To simplify the problem, we take L = 2 and u and d quarks have a equal mass. In this case,η is identified as the U(1) particle and there will not be a mixing between π0 and η.The complex meson field φ contains eight particle excitations σ, η, πa and αa (a = 1, 2, 3):φ = 12(σ + iη) + 12(⃗α + i⃗π) · ⃗τ(4.18)where τ 1,2,3 are the Pauli matrices.

In terms of physical fields, V0, Vm and Vk can be rewrittenas20

V0(φφ†) = −µ22 (σ2 + η2 + ⃗α2 + ⃗π2) + λ18 (σ2 + η2 + ⃗α2 + ⃗π2)2 +λ22 [(σ⃗α + η⃗π)2 + (⃗α × ⃗π)2](4.19)Vm(φ, φ†) = −14meiχ(σ + iη) −14me−iχ(σ −iη)(4.20)Vk(φ, φ†) = −12Kf 2(σ2 −η2 −⃗α2 + ⃗π2) cos θ −K(ση −⃗α · ⃗π) sin θ(4.21)Assuming, for convenience,⟨φ⟩= 12⟨σ + iη⟩= 12veiϕ(v > 0). (4.22)we get, by taking the extremum of V0 + Vm + Vk with respect to v and ϕv2 = 2µ2λ1+ 2mλ1v cos(χ + ϕ) −2Kf 2λ1cos(θ −2ϕ)(4.23)andm sin(χ + ϕ) −Kf 2v sin(θ −2ϕ) = 0.(4.24)Eq.

(4.24) plays a role of the vacuum alignment in the effective theory. If we take ϕ = 0as we wish, (4.24) implies a consistency constraint on χ and θ: They are not separatelyindependent parameters.

They can expressed in terms of the physical parameter ¯θ = θ + 2χassin χ ∼= −Kf 2vm + 2Kf 2v sin ¯θ(4.25)sin θ ∼= −mm + 2Kf 2v sin ¯θ(4.26)where we have assumed that sin χ is very small (<< 1).Rewriting Leff in terms of the shifted field φ →⟨φ⟩+ φ, we getLeff = −¯ψ(̸Ds + 12fv)ψ −14F 2s −(f ¯ψLφψR + h.c.) −Tr(∂µφ∂µφ†) −12(σ, η)M2σηση−12(⃗α,⃗π)M2απ⃗α⃗π−λ1v2 σ(σ2 + η2 + ⃗α2 + ⃗π2) −(4.27)λ2v⃗α · (σ⃗α + η⃗π) −λ18 (σ2 + η2 + ⃗α2 + ⃗π2)2 −λ22 (σ⃗α + η⃗π)2 −λ22 (⃗α × ⃗π)221

where the meson mass matrices are givenM2ση =λ1v2 + mv cos χ−12Kf 2 sin θ−12Kf 2 sin θmv cos χ + 2Kf 2cosθM2απ =λ1v2 + mv cos χ + 2Kf 2cosθ12Kf 2 sin θ12Kf 2 sin θmv cos χ. (4.28)The quark acquires a large constituent massmQ = 12fv ∼= fµ2λ1+ fmλ1v + Kf 3λ1.

(4.29)It is interesting to note that MQ arises from three parts: the spontaneous chiral symme-try breaking (from V0), the explicit chiral symmetry breaking (from Vm) and the instantoninduced symmetry breaking (from Vk). The instanton does spontaneously break chiral sym-metry SU(L)L × SU(L)R [30].

The mass spectrum of mesonic states can be read offfromdiagonalizing (4.28). The mixing probability is proportional to (Kf 2 sin θ)2 = m2 sin2 χwhich is of high order.

It hardly affects the physical massesm2η = mv cos χ + 2Kf 2cosθ,m2π = mv cos χ;m2σ = λ1 + mv cos χ,m2⃗α = λ2v2 + mv cos χ + 2Kf 2 cos θ. (4.30)(4.30) clearly shows how the instanton induced Vk leads to a mass splitting between pionsand the U(1) particle η.

When ¯θ thus θ is small,m2η −m2π = 2Kf 2,(4.31)and in the chiral limit m →0, m2π →0 but m2η →2Kf 2. We conclude that the U(1)problem is solved in the framework of the effective theory if 2Kf 2 is big enough.CP-violating effects originates from the mixing between the scalar and pseudoscalars.To diagonalize the quadratic terms in (4.27), we define the physical meson fields (the primefields)σ = σ′ cos γ + η sin γ,η = −σ′ sin γ + ηcosγ;(4.32)⃗α = ⃗α′ cos γ′ + ⃗π sin γ′,⃗π = −α sin γ′ + ⃗π cos γ′(4.33)22

such that the off-diagonal elements in (4.28) vanish. The mixing angles γ and γ′ are deter-minedγ = Kf 2 sin θm2σ −m2η= 12m2πm2σ −m2η 1 −m2πm2η!¯θ(4.34)γ′ = −Kf 2 sin θm2α −m2π= −12m2πm2α −m2π 1 −m2πm2η!¯θ(4.35)which meet the criteria that the mixing therefore strong CP violation must disappear asm2π →0 or m2η = m2π or ¯θ = 0.

In terms of the physical fields, the CP-violating part of theeffective potential is identified (for simplicity we drop the prime notations)VCP = λ1v2 sin γ η(σ2 + η2 + ⃗α2 + ⃗π2) + λ2v cos γ′ sin(γ −γ′)⃗α · (η⃗α −σ⃗π) +λ2v sin γ cos(γ −γ′)⃗π · (σ⃗α + η⃗π)(4.36)and the Yukawa coupling between quarks and mesons contains CP-violating part tooLyukawa = −12¯ψ(sin γ + iγ5 cos γ)ψη −12¯ψ(sin γ′ + iγ5 cos γ′)⃗τψ · ⃗π. (4.37)The Feynman rules for CP-violating vertices and the typical CP-violating qq →qq amplitudeare shown in Fig.

1.The amplitude of η →2π decays reads from (4.36)A(η →2π) = 14m2πFπ 1 −m2πm2η!¯θ(4.38)where Fπ = v2. (4.38) does not have a direct comparison with the QCD calculation (3.18)and (3.19) where we worked in the case L = 3 and η is one of the would-be goldstone bosons.However, in (4.38), η has been referred to as the U(1) particle.C.

The EDM for the Constituent QuarkThe CP-violating Yukawa coupling in (4.37) results in an important strong CP effect:the EDM of the constituent quark. It can be examined by introducing an external electro-magnetic field Aemµand computing the effective interaction of the type23

µEDM ¯ψγ5σµνψF emµν . (4.39)The coefficient µEDM is defined as the EDM of the quark.

Since (4.39) is not invariantunder the chiral rotation, we have to check the phase of the constituent quark mass mQ.In our convention, mQ is real at tree-level.At higher level, the mass acquires infiniterenormalization. The renormalizability of our model guarantees that the renormalized masswill not develop a γ5-dependent counterpart.

It is still possible that mQ acquires a finiterenormalization which may contain a γ5-part at higher order. But that phase is too smallto cancel (4.39).In the background of EM field, the charged quarks and pions coupling to Aemµthroughthe covariant derivative Demµ−¯ψQ ̸DemQ ψQ −Demµ π+2(4.40)whereDemµ,Q = ∂µ + eQAemµ(4.41)and Q is the electric charge of the particle.

Following Schwinger’s formalism [34] on thederivation of the anomalous magnet moment of electron, we obtain the effective interactionsZd4xLemeff = −Zd4xXQ=u,d¯ψQ(̸DemQ + mQ)ψQ−f 22!Zd4xd4yXQ=u,d¯ψQ(x)eiγ′γ5Sπ0π0SQF (x, y)eiγ′γ5ψQ(y)(4.42)−f 22!Zd4xd4y¯u(x)eiγ′γ5Sπ+π−SdF(x, y)eiγ′γ5u(y)−f 22!Zd4xd4y ¯d(x)eiγ′γ5Sπ+π−SQF (x, y)eiγ′γ5d(y)where Sππ’s and SQF ’s are pion and quark propagators in the background of Aemµ ,Sπ0π0 =1∂2 −m2π,Sπ+π−=1(Demµ )2 −m2π;SQF =1̸DemQ + mQ. (4.43)24

Because e24π << 1, we can expand these propagators perturbatively in eSQF ≠DemQ −mQ(DemQ )2 −m2Q 1 +12eQσµνF emµν(DemQ )2 −m2Q+ · · ·! (4.44)Sπ+π−=1∂2 −m2π 1 + eAemµ ∂µ + e∂µAemµ∂2 −m2π+ · · ·!

(4.45)where the elliptic notation denotes O(e2).The extraction of the effective interaction of(4.39) is done with the aid of Feynman diagrams in Fig. 2.

The contributions from thesecond term in (4.42) correspond to Fig. 2(a), the third to Fig.

2(b) and the fourth to Fig.2(c). Summing them up, we getµuEDM = µdEDM = ef 232π2 sin 2γ′mQ"−231m2Q −m2π+m2Q(m2Q −m2π)2 ln m2Qm2π#.

(4.46)The EDM of neutron is obtained by applying the SU(6) quark model,µneutronEDM= 43µdEDM −13µuEDM ∼=e2mQf 216π2 sin 2γ′ ln m2Qm2π(4.47)where we have used m2Q ≪m2π and γ′ is given in (4.35).V. POSSIBLE SOLUTIONS TO THE STRONG CP PROBLEMIn above, we have studied extensively the measure of strong CP violation and its physicaleffects from viewpoints of QCD and an effective chiral theory.

Jstrong is a product of quarkmasses, ¯θ and the instanton amplitude K(L).It should vanish when any one of themvanishes.The most stringent experiment constraint on Jstrong comes from the EDM ofneutron, which has been measured at a very high precision [35]µneutronEDM< 1.2 × 10−25ecm. (5.1)this impliesJstrong < 10−16GeV 4.

(5.2)At a typical hadron energy scale, one would suspect Jstrong ≃Λ4QCD ≃10−4 ∼10−6GeV 4,enormously larger than the upper limit. This is so-called strong CP problem.

It has puzzledus for more than a decade, ever since the instanton was discovered.25

A. The strong CP Problem or the U(1) Problem?If the instanton is to solve the U(1) problem as we have seen in Sect.

4, the vacuum-to-vacuum amplitude K(L) is related to the mass of the U(1) particle. (5.2) then implies¯θ < 10−10, a very unnatural value since the CP symmetry is violated in weak interactionssince Jweak ̸= 0.

The strong CP problem and the U(1) problem are so closely related that asolution to one actually repels its resolution to another one. In the context of QCD, thereis no theoretical bias to decide which one of them is solved and the other keeps mysterious.Both of them are equally serious in the sense that any solution would be incomplete if itfails to solve both.However, it may be more natural to argue that K(L) is as small as 10−10GeV 2.

In theinstanton computationK(L) ∝¯ρL−4e−8π2g2(¯ρ)(5.3)where ¯ρ is the average density of the instanton gas. The exponential behavior in (5.3) isa standard factor for quantum tunneling and other non-perturbative amplitude.

When theinstanton density is small as required by the validity of the DGA, (5.3) is exponentially smalland can naturally provide a suppression factor of 10−10 while only requiring a reasonablesmall value of αs(¯ρ) =g2(¯ρ)4π≃0.2 ∼0.3.The extreme smallness of K(L) can also beobserved in the large Nc limit [13] where it behaves like e−Nc. If this indeed is true, sin ¯θcan be of order 1.

There is no strong CP problem.Of course, this would leave the U(1) problem unsolved. As is argued by Witten andVeneziano [14], the instanton may not be fully responsible for the mass of the U(1) particlealthough it does break U(1)A symmetry.

The amplitude of the symmetry breaking may befar too small to produce an enough mass for η (L = 2) or η′ (L = 3). They further point outthat based on a reconciliation with the quark model, m2η is of order1Nc in the1Nc expansion.In this case, the mass of the U(1) particle is related to the topological susceptibility in pureYang-Mills theory26

m2η ∼= 4⟨⟨ν2⟩⟩Y MF 2π. (5.4)It is necessary to have a Kogut-Susskind [33] type of a gauge ghost in order to realize thisscenario.

It is not clear whether this is or not a separate solution to the U(1) problem withoutimposing the strong CP violation. But it is worth noting that the strong CP problem inQCD may not be as serious as we thought if we do not insist on a solution to the U(1)problem by the same mechanism.B.

mu = 0 ScenarioWhen mu = 0 thus Jstrong = 0, the strong CP problem is most neatly and elegantlysolved. In the meantime, the U(1) problem can be solved by instanton without resortingto other assumptions.

There is an additional U(1)A symmetry associated with u quark.Thus mu = 0, unlike setting ¯θ = 0, does increase the symmetry of the system and does notviolate ’t Hooft’s naturalness principle. However, that mu = 0 seems to contradict with thephenomenology where mexpu≃5 ∼10MeV [36].However, there is a loophole in this argument [37].

The instanton explicitly breaks U(1)A,as well as U(1)uA associted with the massless u quark if all other light quarks are massive.The instanton is acting as a flavor-changing force, as a result, u quark acquires a radiativemass from other flavors! This is again due to the existence of the zero modes of ̸D in thenontrivial instanton field.

In the presence of a massless fermion, the vacuum tunneling effectis suppressed unless we insert an operator that contains enough grassmann fields to ‘kill’ allthe zero modes. In the ν = ±1 sector, the only operator which survives is ¯uu.

To see howit works, let’s recall the partition function Z(θ) in (3.9). ⟨¯uu⟩is calculated by taking theaverage over space-time⟨¯uu⟩instanton = 1V ⟨Zd4x¯uu(x)⟩= −1Vddmuln Z(¯θ)= −2md · · · mLK(L)(5.5)27

where we have rotated ¯θ to zero as we can when mu = 0. (5.5) implies that U(1)uA sym-metry is broken by instanton.

Of course we would not have the goldstone boson since it isreferred to as an explicit breaking. We should not confuse the condensate ⟨¯uu⟩caused bythe spontaneous symmetry breaking with ⟨¯uu⟩instanton.

The former can be non-zero evenif all quarks are massless while the latter vanishes if d quark mass is zero. The instantoninduced u quark mass can be roughly estimated [31] in the case L = 2 where K(2) is relatedto m2η,minstantonu∼= −παs(¯ρ)CF ¯ρ2⟨¯uu⟩instanton= 43παs(¯ρ)¯ρ2F 2πm2η −m2πm2Qmd ∼= 4 MeV(5.6)where we take ¯ρ ≃( 13ΛQCD)−1, K = −12f2 (m2η −m2π) and f =2mQFπ .

minstantonumust beviewed as an explicit mass because of its proportionality to md. What seems remarkable isthat the order of magnitude of minstantonuis in consistence with the phenomenological value.The massless u quark is still the most favorable solution to the strong CP problem.C.

Peccei-Quinn SymmetryAnother possibility of rendering Jstrong = 0 is that ¯θ = 0 for some dynamical reason.This is realized if the phase of the quark masses θQF D = Pi ϕi is equal to −θQCD. A decadeago, Peccei and Quinn [38] suggested that the strong CP problem may be naturally solved ifone or more quarks acquire the current masses entirely through the Higgs mechanism wherethe lagrangian of quarks and scalars exhibits an adjoint chiral symmetry: the Peccei- Quinnsymmetry.For simplicity, let us examine a toy model of a single quarkLtoy = −¯ψ ̸Dψ −14F 2 + iθF ˜F −(f ¯ψLψRφ + h.c.) −∂µφ∂µφ∗−V0(φ, φ∗)(5.7)whereV0(φ, φ∗) = −µ2φφ∗+ 14λ(φφ∗)2.

(5.8)28

(5.7) is invariant under the PQ symmetryψR →eiαψR,ψL →e−iαψL;φ →e−2iαφ,φ∗→e2iαφ∗. (5.9)The PQ symmetry is quantumly broken by the chiral anomaly, and effectivelyLtoy →Ltoy −2iαF ˜F.

(5.10)Choosing α = θ2 yields ¯θ = 0.The effective potential of the scalar fields can be calculated in a similar way to (4.16)Veff(φ, φ∗) = −µ2φφ∗+ 14λ(φφ∗)2 −Kf ∗e−iθ det φ∗−Kfeiθdetφ(5.11)where K is the instanton amplitude. The last two terms in the effective potential breaksthe PQ symmetry.

The VEV’s of φ and φ∗are found to be⟨fφ⟩= ve−iθ;⟨f ∗φ∗⟩= veiθ(5.12)andv2 = 2µ2|f|2λ+ 2K|f|4λv. (5.13)Thus the fermion mass reads from the Yukawa interaction m = fve−iθ and¯θ = θ + arg⟨fφ⟩= 0.

(5.14)The axion [39] mass is readily derived from (5.11) by diagonalizing the quadratic termsm2axion = 2K|f|2v. (5.15)Unfortunately, we have not been able to discover this particle yet so far.29

VI. CONCLUSIONSWe have studied the measure of CP violation in strong interactions.

It arises from thenontrivial topological structure of Yang-Mills fields, a non-zero vacuum angle ¯θ as well asnonvanishing quark current masses. The instanton dynamics makes most sense in dealingwith the topological gauge configurations where the semiclassical method applies.

It hasbeen shown that the instanton dynamics, as a consistent field theory, automatically satisfiesthe so-called anomalous Ward identity. Crewther’s original complaints on the topologicalsusceptibility and θ-periodicity of the fermion operator are a result of inconsistently handlingthe singularities in some fermion operators.

We conclude that QCD theory itself does notput any constraint on the instanton computation.In the presence of the chiral anomaly, there is no would-be goldstone particle. By study-ing an effective chiral theory, we find that the instanton leads to an explicit U(1)A symmetrybreaking.

If the instanton is to solve the U(1) problem, the measure of the strong CP vio-lation is connected to the mass of the U(1) particle. It may be natural to think that strongCP problem is the side effect of the U(1) problem and both problems cannot be solvedsimultaneously in the context of QCD.However, we point out that the massless u quark scenario to solve the strong CP problemmay not be such a silly idea.

u quark may acquire a mass from d quark through the instantoninteraction in which the fermion zero modes plays an essential role. In any case, with thefailure to observing axions experimentally, the strong CP problem is wide open to newmechanisms.ACKNOWLEDGEMENTI wish to thank Drs.

D.D. Wu and K.S.

Viswanathan for useful discussions.30

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FIGURESFIG. 1.

Feynman rules for η3 and ηπ2 couplings. The CP-violating qq →qq scattering.

Wehave assumed that m2σ ≫m2η, m2α ≫m2π and v = 2Fπ.FIG. 2.

Diagramatic representations of Schwinger’s formulation on the EDM’s for constituentquarks.34

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