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Comments on the Vacuum Orientations in QCD

arXiv:hep-ph/9206251v1 25 Jun 1992SSCLSFU-Preprint-92-5Comments on the Vacuum Orientations in QCDZheng Huang and K.S. ViswanathanDepartment of Physics, Simon Fraser UniversityBurnaby, B.C.

V5A 1S6, CanadaDan-di Wu∗Superconducting Super Collider Laboratory†2550 Beckleymeade Avenue, Dallas, Texas 75237AbstractWe study the QCD vacuum orientation angles in correlation with the strong CPphases. A vacuum alignment equation of the dynamical chiral symmetry breaking isderived based on the anomalous Ward identity.

It is emphasized that a chiral rotationof the quark field causes a change of the vacuum orientation and a change in thedefinition of the light pseudoscalar generators. As an illustration of the idea, η →2πdecays are carefully studied in different chiral frames.

Contrary to the claim in Ref. [7],the θ-term does not directly contribute to the CP-violating processes.May 12, 2019∗On leave of absence from School of Physics, University of Melbourne, Parkville, Vic 3052, Australia.† Operated by the University Research Association, In., for the U.S. Department of Energy under ContractNo.

DE-AC35-89ER40486.1

In QCD lagrangian of strong interactions, there are two possible sources of CP violation:the complex quark mass terms and the θ-term. It has been long realized that they are relatedto each other by chiral transformations associated with the quark fields.

The physical effectsof CP violation only depend on a chiral-rotation invariant ¯θ defined as¯θ = θQCD + θQF D = θQCD +LfXiφi(1)where θQCD is the coefficient of the θ-term, φi is the phase of the ith quark mass term, andLf is the number of light quarks1. However, there is another source of CP-violating angles,the phases of the quark condensates, which arise from dynamical chiral symmetry breaking(DCSB)⟨¯ψiLψiR⟩= −Ci2 eiβi ; ⟨¯ψiRψiL⟩= −Ci2 e−iβi(2)where ψ is the quark field and Ci’s, βi’s are real.

The QCD vacuum orientation is charac-terized by a set of phases of the quark condensates. If the vacuum angle βi ̸= 0 it followsthat ⟨¯ψiiγ5ψi⟩= Ci sin βi ̸= 0 which may also break CP symmetry since ⟨¯ψiγ5ψ⟩is a P-oddand CP-odd quantity.

It has been proven by Vafa and Witten that when ¯θ = 0 and φi = 0for all i’s, the parity symmetry in a vector-like theory such as QCD is not spontaneouslybroken [1] therefore βi = 0 or π [2] for all i’s. When ¯θ ̸= 0, on the other hand, one generallyexpects that the CP-violating interactions in the lagrangian may result in a CP-asymmetricphysical vacuum.

The purpose of the present paper is to study the vacuum orientation in thepresence of strong CP violation and its potential effects on CP-violating processes in stronginteractions. We find that the phases of quark condensates can be completely determined asfunctions of ¯θ and φi’s via a vacuum alignment equation.

Thus βi’s are not spontaneouslygenerated either even when the CP symmetry is explicitly violated by ¯θ ̸= 0.Obviously, the quark condensates (2) cannot be referred to as fundamental parameters ofthe theory since they are subject to chiral transformations. In fact βi’s can be set to any val-ues if we make appropriate chiral transformations for the quark fields.

Such transformations1The inclusion of heavy quarks will not change our discussion significantly if they are in the normal phase.Otherwise see Ref. [4].2

also change the phases of the quark masses, as well as the coefficient of the θ-term because ofthe chiral anomaly. But what is important is the correlation between the vacuum orientationand the distribution of the strong CP phases among θ-term and quark mass terms, which isto be determined by the vacuum alignment.

The effective CP-violating interactions in lowenergy hadron physics (for instance, in current algebra) highly depend on this correlatingfeature. As we shall see below, the sole ¯θ-dependence of the strong CP effects is proven onlywhen the orientations of the vacuum are properly considered.

In addition, it is of interest tostudy DCSB in the presence of strong CP violation in its own right.One way of relating the phases of the quark condensates with θQCD and φi’s is to considerthe so-called anomalous Ward identity [3]12∂µJ(i)5µ=F ˜F + imieiφi ¯ψiLψiR −imie−iφi ¯ψiRψiL(3)(i = 1, 2, . .

. , Lf)where J(i)5µ= ¯ψiγµγ5ψi, F ˜F =g232π2ǫµνρσF µνF ρσ and F µν the non-abelian gauge field strengthtensor.

Taking the vacuum expectation values (VEV) on both sides of (3) yields [4]DF ˜FE= −imi[eiφi D ¯ψiLψiRE−e−iφi D ¯ψiRψiLE](4a)= −miCi sin(φi + βi)(i = 1, 2, · · ·, Lf). (4b)In deriving (4a) we have assumed that the VEV’s of the divergence of the gauge invariantcurrent vanish.

Eq. (4b) is the master equation of this paper.

It is important to point outthat if the DCSB does not occur, (4a) would vanish identically and there is no constraint onthose phases. Indeed even though the quark condensate can be non-zero due to the explicitchiral symmetry breaking (ECSB) i. e. the quark current masses, it does not contribute to(4a) because it possesses a phase opposite to the phase of the quark mass φi and rendersthe RHS of (4a) zero.

This can be easily seen by taking the free-quark limit in which thecondensate is calculated asD ¯ψiLψiRE= 12Zd4k(2π)4 Tr1 + γ5̸ k −mieiφiγ5 = −miΛ2(mi)e−iφi(5)3

where Λ2 is real. The substitution of (5) into (4a) yields ⟨F ˜F⟩= 0.

Therefore, Ci’s in (4a)should be understood as the purely dynamical condensates originating from DCSB. Thekinematical part of the condensates that is induced by the ECSB and has a phase −φi hasbeen subtracted out in (4a).

It is the DSCB combining with the topological structure ofQCD Yang-Mills fields (the instanton effect) that makes the strong CP phases non-trivialand relates them to each other.Eq. (4b) is not immediately useful to us since it has an unknown quantity ⟨F ˜F⟩.Itinvolves no quark fields thus is independent of the chiral transformation.

It is conceivablethat ⟨F ˜F⟩is solely a function of ¯θ (not of φi’s and βi’s separately). A rigorous proof can bemade by summing over instanton configurations in QCD θ vacua.

For simplicity, considerQCD with a single quark field ψ. The VEV’s of F ˜F is given by [5]DF ˜FE=1V TZd4xF ˜F(6)=1V T1NXν=0,±1,···ei¯θννZ[dAµ]ν det(i ̸Dν + im) exp(−Zd4xFF)where N is the normalization factor, V T is the volume of Euclidean space-time, and ν isthe winding number of the instanton field configuration, and the fermion determinant resultsfrom the integration over the quark field.

We have made an appropriate chiral transformationsuch that the quark mass is real and θQCD = ¯θ (we can always do so because the generatingfunctional is invariant under the redefinition of integral variables). It is shown that whenν > 0 (< 0) i ̸Dν has |ν| zero modes with negative (positive) chirality [6].

We thus obtainDF ˜FE= mA1(m2) −i2 (ei¯θ −e−i¯θ)+m2A2(m2) −i2 (ei2¯θ −e−i2¯θ) + · · · + m|ν|A|ν|(m2) −i2 (ei|ν|¯θ −e−i|ν|¯θ) + · · ·= mA1(m2) sin ¯θ + m2A2(m2) sin 2¯θ + · · · = K(m, θ) sin ¯θ(7)where A|ν|(m2)’s are given in Euclidean spaceA|ν|(m2) =1V TN e−|ν| 8π2g2Z[dAµ]νYλr>0[λ2r(A) + m2] exp[−Zd4xFF]. (8)4

Here λr(A)’s are non-zero eigenvalues of i ̸Dν. Clearly A|ν|(m2)’s are some real functionsof m2 and do not vanish as m →0.If ¯θ is small as it must be, ⟨F ˜F⟩≃K(m)¯θ =K(m)θQCD + K(m)θQF D.Combining (7) with (4b), we derive the so-called vacuum alignment equation (VAE) [4, 8],which determines the orientation of the QCD vacuum in the presence of strong CP violationK(m)¯θ = miCi(φi + βi) + O(m2; ¯θ2)(9)(i = 1, 2, · · · , Lf).Eq.

(9) has proven that βi’s are not spontaneously generated even when ¯θ ̸= 0. The conclusionof Vafa and Witten’s theorem [1] can be extended to the case where parity symmetry isexplicitly violated.

A similar result has been worked out previously [8] from different pointsof view.If mi’s vanish, βi’s can be arbitrary.This is referred to as the degeneracy ofQCD vacua when the ECSB is absent. Any vacuum characterized by a set of the vacuumangles βi’s is as good as others and the orientation of the DCSB is arbitrary.

However, theimportance of (9) is that when the ECSB is turned on, the ground state must align with itin such a way that (9) is satisfied. Though both φi and βi are not physical parameters andcan be changed through chiral rotations, their sum is uniquely determined by the physicalparameter ¯θ.

When one is chosen the other is completely determined through making thevacuum alignment. As is emphasized by Dashen [9], a misaligned vacuum, whose orientationangles do not satisfy (9), may cause an inconsistency such as the goldstone bosons (pions)acquiring negative mass squares.Once the DCSB and the ECSB align with each other, an absolute rotation of the wholesystem is of no concern.

Thus a chiral transformation is allowed only if the correspondingchange of the vacuum orientation has been taken into account. We can have two ways to makethe vacuum alignment.

We may choose one particular vacuum, for example, by requiring thequark condensate to be real βi = 0 (i = 1, 2, · · ·, Lf) and ask what perturbation (the ECSB)is aligned with it. Recalling that Ci’s are dynamical condensates and thus Ci = Cj = C5

(i, j = 1, 2, · · · , Lf), we obtain by solving (9) for φi’s, to O(m; ¯θ)(A)φi = K(m)mi¯θ ; βi = 0(i = 1, 2, · · · , Lf)θQF D =Xiφi = K(m)¯m¯θ;θQCD = ¯θ −θQF D = K(m) −¯m¯m¯θ(10)where ¯m = (Pi1mi)−1 and the CP-violating lagrangianLCP(A) = −Ximiφi ¯ψiiγ5ψi + θQCDF ˜F = −K(m)¯m¯θ ¯ψiγ5Iψ + K(m) −¯m¯m¯θF ˜F(11)where I is an identity matrix. We shall call the solution (10) basis (A).

Another way isto assume a certain pattern of the ECSB and to ask which one of the degenerate vacuacorresponds to the perturbation. For example, we may choose the quark mass terms realφi = 0 (i = 1, 2, · · ·, Lf) and determine the vacuum angle βi’s.

Again, from (9) we have(B)φi = 0;βi = K(m)mi¯θ(i = 1, 2, · · · , Lf)θQF D =Xiφi = 0;θQCD = ¯θ(12)andLCP(B) = ¯θF ˜F. (13)Solution (12) is to be called basis (B).

We would like to emphasize again that by performingthe chiral rotation on quark fields one has switched the strong CP phases among the θ-termand quark mass terms, and obtained different lagrangians, each of which corresponds to acertain vacuum orientation. When calculating the strong CP effects we must take this intoconsideration to assure the correct result.As an illustration, we compute the CP-violating η →2π decays in two bases with a given¯θ.

In basis (A) where the condensates are real, we apply the soft-pion theorem to extractingη and π’s2A(η →π+π−) =Dπ+π−LCP(A) ηE2We have worked in the context of SU(3)L × SU(3)R where η and π’s are all light pseudoscalars.6

= −K(m)¯m¯θ(−iFπ)3 D[Q85, [Q+5 , [Q−5 , ¯ψiγ5Iψ]E(14)≃−K(m)¯m¯θ 1FπD ¯ψ{ λ82 , { λ+2 , { λ−2 , I}ψE= K(m)¯m¯θ 1F 3π2√3Cwhere the pion decay constant Fπ ≈93MeV and we have used [Qa5, F ˜F] = 0. The brokengenerators of SU(3)A corresponding to light pseudoscalars are given byQa5 =Zd3xψ†γ5λa2 ψ(x)(a = 1, 2, · · ·, 8)(15)where λa’s are Gell-Mann matrices and λ± = 1/√2(λ1 ∓iλ2).

(14) has been first derivedby Crewther, Di Vicchia, Veneziano and Witten (CDVW) [10]. However, there have beendoubts about the calculation since it does not explicitly exhibit the use of the topologicalnon-triviality of the θ-vacuum.

More concretely, one may shift the strong CP phases fromθQF D to θQCD through chiral rotations and computes the amplitude, as one does in (14),Dπ+π−LCP(B) ηE= ¯θ(−iFπ)Dπ−[Q−5 , F ˜F] ηE(16)which is zero if one imposes the canonical commutation relation by which Qa5 commutes withgauge fields. This contradiction has triggered a serious doubt on whether or not the strongCP phases lead to any physical effects at all [11].We believe that this concern is redundant.

The vacuum alignment equation (VAE) hasincorporated the non-perturbative features of QCD vacuum into the game. Both LCP(A) andLCP(B) are solutions of the VAE and should, if one does things correctly, result in the sameconclusion.

In basis (B) the quark masses are real but the condensates are complex. Thevacuum does not respect CP symmetry.

In this case even though LCP(B) does not contribute tothe amplitude as shown in (16), the CP conserving part of the lagrangian may do. Moreover,when the quarks have non-degenerate masses (mass splitting), the condensates are of theformD ¯ψLψRE= −C2 β ≡−C2 (I + iδ)(17a)withβ =eiβueiβd...; δ ≃βuβd...(17b)7

if βi’s are small. Apparently (17a) is not invariant under SU(Lf)V transformations if βi’sare not all equal.

In other words, the vector charges defined as generators of SU(Lf)V donot annihilate the vacuum completely orQa |0⟩̸= 0. (18)Clearly, the subgroup of SU(Lf)L×SU(Lf)R which leaves the vacuum invariant must satisfyU+L βUR = βorUR = β−1ULβ(19)where UL and UR are left and right unitary representations of SU(Lf).

The broken gen-erators, which excite the glodstone bosons known as pions, are those of the coset of theunbroken group. From (19) it is easy to understand that the broken group is not SU(Lf)Aany more but to be rotated to β−1SU(Lf)Aβ generated by δ.

The pion generators, denotedby ˜Qa5, are thus˜Qa5 =Zd3x{ψ†γ5λa2 ψ(x)+ψ†[ λa2 , iδ]ψ(x)} + O(δ2),(20)i. e. , the pions are mixing of P-odd and P-even components.Now that LCP(B) = ¯θF ˜F has no contribution to the amplitude, we haveA(η →π+π−) =Dπ+π−−¯ψmψ ηE= −(−iFπ)3 D[ ˜Q85, [ ˜Q+5 , [ ˜Q−5 , ¯ψmψ]E(21)= −iF 3π{D ¯ψ[[ λ82 , iδ], { λ+2 , { λ−2 , m}}]ψE+D ¯ψ{ λ82 , [[ λ+2 iδ], { λ−2 , m}]}ψE+D ¯ψ{ λ82 , { λ+2 , [[ λ−2 , iδ], m]}}ψE−D ¯ψγ5{ λ82 , { λ+2 , { λ−2 , m}}}ψE}where m is the diagonal mass matrix which is real in this basis. The first three terms inparenthesis come from the modification of the pion generators and the last term reflects thecomplexity of the condensates which is absent in basis (A).

They are of the same order of mand ¯θ. Manipulations of these commutators yield, to O(m; ¯θ)A(η →π+π−) = −iF 3π{ i√3(βu −βd)(−mu ⟨¯uu⟩+ mdD ¯ddE)8

−1√3(mu + md)(⟨¯uγ5u⟩+D ¯dγ5dE)} = K(m)¯m¯θ 1F 3π2√3C. (22)In deriving the final step of (22) we have substituted in ⟨¯ψiψi⟩= −C cos βi ≃−C and⟨¯ψiiγ5ψi⟩= C sin βi ≃Cβi and solution (12).

We therefore confirm that CDVW’s result isindependent of chiral frames.We conclude that the study of the vacuum orientation of the dynamical chiral symmetrybreaking provides us an improved understanding of strong CP violation. It has been shownthat η →2π decay occurs if ¯θ is non-zero.

More precise experiment measuremental on thedecay rate is encouraged to constrain ¯θ .Huang would like to thank F. Gilman for arranging him a visit to SSC Lab when thiswork was planned. A stimulation from S. Weinberg is gratefully acknowledged.9

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