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Light Quark Masses and Quarkonium Decays

arXiv:hep-ph/9206247v1 24 Jun 1992UCSD/PTH 92/06Light Quark Masses and Quarkonium DecaysKiwoon ChoiDepartment of PhysicsUniversity of California at San Diego9500 Gilman DriveLa Jolla, CA 92093-0319, USAAbstractAfter discussing the intrinsic ambiguity in determining the light quarkmass ratio mu/md, we reexamine the recent proposal that this ambiguitycan be resolved by applying the QCD multipole expansion for the heavyquarkonium decays. It is observed that, due to instanton effects, somematrix elements which have been ignored in previous works can give asignificant contribution to the decay amplitudes, which results in a largeuncertainty in the value of mu/md deduced from quarkonium phenomenol-ogy.

This uncertainty can be resolved only by a QCD calculation of somesecond order coefficients in the chiral expansion of the decay amplitudes.

It has been observed by a number of authors [1 −5] that second order correctionsin chiral perturbation theory can significantly affect the estimate of the light quarkmasses. In particular, it was pointed out that the determination of mu/md suffersfrom a large uncertainty due to the instanton-induced mass renormalization [1, 3]:M →¯M(ω) ≡M + ωMI(1)where the real matrix M = diag(mu, md, ms) denotes the light quark masses inthe QCD lagrangian, MI ≡14πf (detM†)(M†)−1 =14πf (mdms, mums, mumd) is theinstanton-induced second order mass with the pion decay constant f ≃93 MeV, andω is a dimensionless parameter of order unity.

Most of the previous analyses on Mdo not distinguish M from ¯M, and thus the corresponding results can be interpretedas those on ¯M for an arbitrary value of ω, which leads to a large uncertainty in theextracted value of mu/md.Recently it was argued that the above mentioned difficulty can be overcome bynoting that the instanton-induced mass MI is distinguished from the bare mass Mthrough its θ-dependence where θ denotes the CP violating QCD vacuum angle [6]. Ifone keeps the θ-dependence explicitly, MI always appears with the phase eiθ due to thewinding number of instantons.

Note that M and eiθMI have the same transformationunder the QCD chiral symmetry SU(3)L × SU(3)R × U(1)A under whichM →eiαLMR†,θ →θ + 3α,(2)where L ∈SU(3)L, R ∈SU(3)R, and α generates the anomalous U(1)A rotation.Then one may be able to measure M directly, not ¯M involving an arbitrary unknownparameter ω, by probing the θ-dependence of the QCD dynamics. Among quantitiesthat probe the θ-dependence, the matrix elementsDφ|G ˜G|0E(φ = π0 or η) wereconsidered in ref.

[6].It was then argued that the ratio RA ≡⟨π0|G ˜G|0⟩⟨η|G ˜G|0⟩can bereliably determined by applying the QCD multipole expansion for the quarkoniumdecays ψ′ →J/ψ + φ, which would allow us to precisely determine mu/md.In2

this paper, we wish to reexamine whether the QCD multipole expansion appliedfor the quarkonium decays can provide a way to determine mu/md without doingany nonperturbative QCD calculation. Our result then confirms the conclusion ofref.

[7], viz. in order to precisely determine mu/md, one needs to calculate a QCDmatrix element which receives a potentially large instanton contribution.

Since sucha calculation is not available at present, a rather wide range of mu/md, includingzero, should be considered as being consistent with our present knowledge of QCD.To proceed, let us briefly review the points that were discussed in refs. [3, 6, 7].In order to extract information on M, one usually considers measurable quantitieswhose M-dependence can be deduced from an effective lagrangian of hadron fields.This is not a severe limitation since it is hard to imagine a measurable quantitywhich may provide useful information on M through its M-dependence but can notbe described by any hadronic effective lagrangian.

To satisfy the Ward identities ofthe QCD chiral symmetry, the effective lagrangian is required to be invariant underthe chiral transformations of the involved hadrons and also those of M and θ givenin eq. (2).

Since M and eiθMI have the same chiral transformation property, forany term in the effective lagrangian which is first order in M, e.g. O1 = ai ⟨MΩi⟩,there exists also the second order term O2 = bieiθ ⟨MIΩi⟩, where ⟨Z⟩= tr(Z), aiand bi are chiral coefficients which are calculable within QCD, and Ωi is a genericlocal functional of hadron fields which is transformed under the chiral symmetry asM−1.

This can be understood within QCD by noting that eiθMI corresponds to theeffective current mass induced by instantons [1, 3]. For any QCD diagram whichcontains an insertion of M and thus gives a contribution to O1, one can replace Mby the instanton-induced effective mass eiθMI to make the new diagram contributeto O2.

This means that the M-dependent part of the effective lagrangian can alwaysbe written asLeff(θ) ⊃Xihai ⟨MΩi⟩+ bieiθ ⟨MIΩi⟩+ ...i,(3)3

where the ellipsis denotes other possible higher order terms.To be invariant under the anomalous U(1)A symmetry, the effective lagrangian isθ-dependent in general. One may then expand the θ-dependent effective lagrangianaround θ = 0:Leff(θ) =n=∞Xn=0θnL(n)eff.

(4)Clearly L(0)eff ≡Leff(θ = 0) describes the normal CP conserving strong interactionswhile the terms of nonzero n describe the θ-dependence of the QCD dynamics, includ-ing the CP violation due to a nonzero θ. In view of the arguments leading to eq.

(3),for any L(0)eff that includes the corrections of O(M2), one can define a transformationof the form:M →¯M ≡M + ωMI,bi →bi −ωai,(5)under which L(0)eff is invariant1.The above transformation mixes the instanton-induced mass MI with the bare mass M, and thus will be called as the instantontransformation (IT) in the following discussion. Obviously L(n)eff (n ̸= 0) in eq.

(4)are not invariant under the IT. This is nothing but to mean that L(0)eff does not distin-guish the instanton-induced mass from the bare one, while L(n)eff (n ̸= 0) distinguishsince they arise from the θ-dependence.

Based on this observation, it was stated inref. [3] that the normal CP conserving strong interactions which are described byL(0)eff always concern the effective mass ¯M with an unspecified value of ω, while theCP violating amplitudes or the nonderivative axion couplings probe directly the baremass M. (Note that up to a small mixing with mesons, θ corresponds to the con-stant mode of axion in axion models.) For instance, if det(M) = 0, then θ can berotated away (even in the case without axion) and axion becomes massless althoughdet( ¯M) ̸= 0.Although L(n)eff’s (n ̸= 0) in eq.

(4) might be useful also, one usually considers only1 Throughout this paper, whenever we say about the IT, it is assumed that the corrections ofO(M 2) are included, while those of O(M 3) are ignored.4

L(0)eff in extracting information on M from experimental data. Of course the reason isthat it is quite nontrivial to find a link between the available experimental data andthe terms of nonzero n. The IT was defined as acting on M and the second orderchiral coefficients bi’s.

Since L(0)eff is invariant, all measurable quantities described byL(0)eff, i.e. expressed in terms of M and other parameters that appear in L(0)eff, arealso invariant under the IT.

This can be understood also by observing that the ITcan be considered as a kind of renormalization group (RG) transformation associatedwith the instanton-induced mass renormalization [7]. In taking this specific instantoneffects into account, one can use the following scheme.

For θ which is fixed to bezero, the contribution to L(0)eff from instantons with size ρ ≤µ−1Iis taken into accountby redefining the chiral expansion parameter as M →M + ωMI, while that fromlarger instantons (ρ ≥µ−1I ) appears in the second order chiral coefficients bi. In thisscheme, the transformation parameter ω of the IT, being naturally of order unity, canbe identified as ω(µI) =R ∞µIdµµ D(µ), where D(µ) denotes an appropriately normalizeddimensionless instanton density.

Then the IT corresponds to a renormalization grouptransformation2 changing µI, and all measurable quantities deduced from L(0)eff shouldbe independent of our choice of µI, i.e. are invariant under the IT.As was argued previously [7], if we restrict ourselves to L(0)eff, the invariance ofL(0)eff under the IT prevents us from precisely determining mu/md.All equationsdeduced from L(0)eff are covariant under the IT.

As a result, any quantity which issensitive to the IT, i.e. its variation under the IT for ω = O(1) can be comparable tothe expected central value, can not be fixed by the invariant experimental data.

(Ofcourse the invariant combination of such quantities can be fixed.) One can not avoidan uncertainty whose size is characterized by the variation under the IT.

For mu/md,2 In considering the IT as a RG transformation, it must be noted that the scale µI is introducedonly for the instanton-induced mass renormalization, but not for other kind of instanton effects.This isolation of the instanton-induced mass renormalization from other instanton effects can beeasily achieved in the dilute instanton gas approximation. Then the conclusion that measurablequantities deduced from L(0)eff are invariant under the IT can be understood by restricting ourselvesto the effects of relatively small instantons for which the dilute gas approximation is valid.5

its variation under the IT is roughly ωms/4πf which can be as large as about 1/2,implying a rather large uncertainty.To be more specific, let us consider the case of using the pseudoscalar mesonmasses as experimental input. The relevant part of the effective lagrangian is [8]L(0)eff ⊃14f 2 Dχ†U + χU†E+ L6Dχ†U + χU†E2+ L7Dχ†U −χU†E2 + L8Dχ†Uχ†U + χU†χU†E,(6)where χ = 2B0M for B0 = −⟨0|¯uu|0⟩/f 2 defined at chiral limit, and the SU(3)-valued U denotes the pseudoscalar meson octet.

For the above terms, the IT of eq. (5) can be written as [2, 5]:mu →mu + ωmdms4πf(cyclic in u, d, s),Li →Li −hiωf128πB0,(7)where h6 = h7 = 1 and h8 = −2.

Note that ms can be considered to be invariantsince its variation is negligibly small compared to ms for ω of order unity. In refs.

[2, 8], it was found that3m2sm2d −m2u≃(M4K/M2π)(M2K0 −M2K+ + M2π+ −M2π0)−1,md + mumd −mu≃(1 + △M)2M2π(M2K0 −M2K+ + M2π+ −M2π0)−1,(8)where △M = −32L7B0ms/f 2 −c for c ≃0.33 = O(ms/4πf). These equations do notfix mu/md, but give only a curve on the plane of (mu/md, λ) where λ = ms/md forthe first equation and λ = L7 for the second.

Note that these curves are parametrizedby ω, and the uncertainties in mu/md and λ are given by their variations under theIT for ω = O(1). We will encounter the same situation even when other kind ofmeasurable quantities, e.g.

the baryon masses and the decay amplitudes for η →3π,ψ′ →J/ψ + π0(η), are used in the context of an appropriate form of L(0)eff.3 Throughout this paper, we will use md/ms ≪ms/4πf, and thus ignore the corrections sup-pressed by either md/ms or md/4πf, while keeping only those of O(ms/4πf).6

A definite value of mu/md would be obtained if one can choose a specific valueof λ, but it is possible only through a QCD calculation of λ since λ is sensitive tothe IT and thus can not be fixed by experimental data alone. Such an attempt wasmade recently by Leutwyler [5] for λ = L7.

By invoking η′-dominance, it was arguedthat L7 falls into a rather narrow range of negative values, thus ruling out mu = 0in view of the second equation in (8). However it has been pointed out later thatinstantons significantly suppress the negative η′-contribution to L7, while enhancingthe positive contribution from the pseudoscalar octet resonances [7].

This would makeη′-dominance for L7 not valid, and thus results in a large uncertainty in the value ofL7, allowing mu = 0. In fact, any quantity which is sensitive to the IT receives apotentially large contribution from instantons.

This makes the QCD calculation of λand the precise determination of mu/md even more nontrivial.So far, we have argued that there exists an intrinsic ambiguity in determiningmu/md by usual manner.This ambiguity is due to the invariance of L(0)eff underthe IT, and as was pointed out in ref. [6], might be resolved by including the non-invariant terms L(n)eff (n ̸= 0) in the analysis.

In QCD, one can probe the θ-dependencethrough the matrix elements involving the insertion of the two gluon operator G ˜G,e.g.,Dφ|G ˜G|0E(φ = π0 or η) andD0|(G ˜G)(G ˜G)|0E. The hadronic realizations of suchmatrix elements contain L(n)eff (n ̸= 0) in general.

In axion models,Dφ|G ˜G|0Edescribesthe axion-meson mixing while the other matrix element is related to the axion mass.As a result, physical processes involving axion may provide information on thesematrix elements, and thus on the bare quark mass M. However it is totally unclearwhether the normal strong interaction data can also provide any information on thesematrix elements.In ref. [6], it was observed that the matrix elementDφ|G ˜G|0Eappears in the QCD multipole expansion applied for the heavy quarkonium decaysΦ′ →Φ + φ (Φ = J/ψ or Υ).The major purpose of this paper is to examinewhether this allows us to determine mu/md without doing any nonperturbative QCDcalculation.7

The quarkonium decays Φ′ →Φ + φ can be described by the effective lagrangianof eq. (3) with Ωi’s containing the SU(3)-singlet heavy quarkonium fields Φ and Φ′together with φ.

The resulting decay amplitudes H(φ) (for Φ′ at rest) can be writtenas:H(φ) = ǫijkΦ′iΦjpk [ˆx ⟨Mφ⟩+ ˆγ ⟨MIφ⟩+ ...] ,(9)where Φ′i and Φi denote the spin vectors of Φ′ and Φ respectively, pi is the momentumof φ = φaλa. Clearly the above amplitudes must be invariant under the IT of eq.

(5),acting on M and also on ˆγ which is proportional to an appropriate combination of bi’s.As was discussed by Voloshin and Zakharov (VZ) [9], the QCD multipole expansionwhose expansion parameter is the inverse of the heavy quark mass provides anotherexpression for H(φ):H(φ) = ǫijkΦ′iΦjXk(φ) W0,(10)where W0 depends on the quarkonium wavefunctions, andXi(φ) ≡Dφ|g2EakDkBai |0E≡piX(φ). (11)At the leading order in the multipole expansion, the quarkonium wavefunctions canbe considered to be independent of M, and thus W0 is invariant under the IT.

Thismeans that X(φ) which is written asX(φ) = x ⟨Mφ⟩+ γ ⟨MIφ⟩+ ...,(12)is invariant under the IT of eq. (5), implying that γ ≡ˆγ/W0 is transformed asγ →γ −ωx.

(13)In any case, one can defineAi(φ) ≡Dφ|g2∂k(EakBai )|0E= piA(φ),Bi(φ) ≡−Dφ|g2Bai DkEak|0E= piB(φ),(14)8

so thatX(φ) = A(φ) + B(φ). (15)Note that A(φ) can be written as A(φ) =i12Dφ|g2G ˜G|0Ewhere G ˜G ≡Gaµν ˜Gaµν.In the attempt to determine mu/md using the quarkonium decays, the quantities ofinterests are RX ≡X(π0)/X(η) and RA ≡A(π0)/A(η).

Then RX can be fixed bythe quarkonium decay data, while it is necessary to fix RA to determine mu/md. Infact RA = RX at the leading order in M, however we need to include the correctionsof O(M2) for a meaningful determination of mu/md.

In ref. [6], following ref.

[9],it was simply assumed that |B(φ)| ≪|A(φ)|, which would imply RX ≃RA even atO(M2). Here we first argue that there is no reason for this assumption to be viable,and later show how the instanton-induced mass renormalization promotes RA to begreater than RX.The statement that B(φ) may be significantly smaller than A(φ) was first madeby VZ [9] who observed that A(φ) = O(√Nc), and thus is enhanced by one powerof Nc with respect to the naive Nc-counting.

This enhancement is due to the η0-polewhere the SU(3)-singlet η0 denotes η′ at chiral limit. Roughly we haveDφ|g2G ˜G|0E≃Dη0|g2G ˜G|0E× M2φη0 ×1M2η0,(16)where the η0-pole (≡1/M2η0) is O(Nc).

Note that the mass-squared mixing M2φη0between η0 and φ is suppressed by the light quark mass M, but is O(1) in the Nc-counting, andDη0|g2G ˜G|0E= O(1/√Nc) obeys the naive Nc-counting rule. In fact,the same enhancement can occur also for B(φ) if ⟨η0|Bai DkEak|0⟩is nonvanishing.Then B(φ) can be equally important as A(φ) even in the large Nc-limit.

If B(φ)does not receive any contribution from the intermediate η0 and thus is O(1/√Nc), wewould have |B(φ)/A(φ)| ≃M2η′/Λ2, where Λ denotes a typical hadronic scale which isO(1) in the Nc-counting. Clearly then X(φ) will be dominated by A(φ) in the largeNc limit.

However in the real case of Nc = 3, this is not necessarily true since Mη′ islarge enough to be comparable to Λ. Again B(φ) can be equally important as A(φ).9

VZ noted also that the equation of motion DkEak = igq† λa2 q gives B(φ) an ad-ditional power of the QCD coupling constant g. However it is hard to imagine thatthis extra g means a real suppression of B(φ) compared to A(φ). First of all, g isessentially of order one, even for the renormalization point above mb, although theloop suppression factor αs/4π = g2/16π2 ≪1.

Note that the use of the equation ofmotion has nothing to do with the perturbative QCD loop expansion. Furthermore ifwe consider the valence quark contribution (in the sense of the parton model) to thevacuum to meson matrix element of an n-gluon operator, it would include a factorgn.

Before using the equation of motion, both A(φ) and B(φ) involve two-gluon op-erators. The equation of motion changes the two-gluon operator in B(φ) to Bai q†λaqwhich includes only one gluon field.

Then at least for the valence quark contribution,B(φ) is not higher order in g compared to A(φ) since the matrix element of Bai q†λaqwill be enhanced by g−1 compared to the matrix elements of two-gluon operators.We have argued that there is no a priori reason to expect that B(φ) is signif-icantly smaller than A(φ).One must include B(φ) in the analysis, and then RAcan be significantly different from RX at O(M2). Among the O(M2) corrections, theinstanton-induced mass renormalization is of particular importance for the determina-tion of mu/md.

Thus from now on, we will concentrate on how the instanton-inducedmass renormalization affects A(φ) and B(φ), so that promotes RA to be greater thanRX. For this purpose, we study the IT of the second order chiral coefficients α andβ that appear in the following chiral expansion4:A(φ) = a ⟨Mφ⟩+ α ⟨MIφ⟩+ ...,B(φ) = b ⟨Mφ⟩+ β ⟨MIφ⟩+ ...,(17)where the ellipses denote other possible second order terms.

Note that eqs. (12)and (15) imply that A(φ) and B(φ) can be expanded as above although these matrixelements can not be evaluated by using L(0)eff alone.4 Here we ignore the electromagnetic effects which were shown to be negligibly small [12].10

In order to derive the IT of α, let us express the physical meson fields π0 and ηin terms of φ3 and φ8 as5: π0 = φ3 + ǫφ8,η = φ8 −ǫφ3. The mixing parameter ǫcan be written asǫ =√34md −mums"1 + ms4πf κ#,(18)where κ is a dimensionless parameter of O(1) which represents the size of the O(M2)-corrections.

Clearly ǫ should be invariant under the IT since it describes the physicalmeson fields in terms of the original wavefunctions φ3 and φ8 which are untouched bythe IT. This then gives the following IT of κ:κ →κ + ω.

(19)In fact, using the chiral lagrangian, ǫ was evaluated as [6]:κ = −128πB0f(L8 + 3L7) + chiral logs,(20)assuring the above IT of κ (see the IT of Li’s in eq. (7)).Using eqs.

(17) and (18), we can obtainA(π0) = 32(mu −md)a"1 + ms4πf (κ3 −2α3a + ...)#,(21)while the anomalous Ward identity∂µ(¯qγµγ5M−1q) =g216π2tr(M−1)Gaµν ˜Gaµν + 2¯qiγ5qgivesA(π0) = 4iπ23mu −mdmu + mdfπM2π"1 + O( md4πf )#,(22)where fπ and Mπ denote the decay constant and the mass of the physical π0 respec-tively. Here and in what follows, ellipsis always denotes a dimensionless coefficientof O(1) which is untouched by the IT.

Then comparing (22) with (21) together with5In fact, due to the SU(3) breaking, the π0-φ8 mixing can be different from the η-φ3 mixing atO(M 2). However this difference does not affect our analysis of the IT, and thus will be ignored.11

the expressions of fπ and Mπ given in ref. [8], we finda=8iπ2B0f9,αa=−64πB0f(L8 + 3L7 + 3L6) + ...,(23)which yield the following ITα →α + 2ωa,β →β −ω(b + 3a).

(24)Here the IT of β is derived from that of α and γ, using x = a + b and γ = α + β.We are now ready to see how the instanton-induced mass renormalization affectsRA and RX, and what is still necessary to determine mu/md using the QCD multipoleexpansion applied for the quarknonium decays. Using eqs.

(12), (17), and (18), wecan obtainRX≡X(π0)X(η) = 3√34md −mums"1 + ms4πf ξX#RA≡A(π0)A(η) = 3√34md −mums"1 + ms4πf ξA#,(25)where ξX = κ/3 −2γ/3x + ...,ξA = κ/3 −2α/3a + ... are dimensionless coefficientsof O(1). Then the IT of α, γ, and κ derived so far gives the following IT:ξX →ξX + ω,ξA →ξA −ω.

(26)We have argued that the IT can be interpreted as a kind of RG transformationchanging the scale µI that appears associated with the instanton-induced mass renor-malization. With this interpretation, the IT of ξA,X for a maximal value of the trans-formation parameter, i.e.

ω = ωmax, represents the negative of the full instantoncontribution to ξA,X. Then ξA receives a positive contribution ωmax from instantons,while ξX receives a negative one with the same size.

Although its precise value isquite sensitive to the unknown nonperturbative QCD dynamics, it has been observedthat ωmax can be as large as 2 ∼3 in the semiclassical instanton gas approximation12

[1, 3]. Then although not very reliable, for such a large ωmax, it is more likely thatRA is significantly greater than RX.Applying the QCD multipole expansion for the measured quarkonium decaywidths Γ(ψ′ →Jψ + φ) (φ = π0 or η), one finds RX ≃4.3 × 10−2 [6, 10, 11].

Alsoin ref. [6], the size of ˆRA ≡(md + mu)RA/(md −mu) was estimated within the sec-ond order chiral perturbation theory of the light pseudoscalar mesons, which yieldsˆRA ≃8 × 10−2.

Note that both RX andˆRA are invariant under the IT, and thuscan be fixed by experimental data. If we assume as in ref.

[6] that |B(φ)| ≪|A(φ)|which implies RA ≃RX or equivalently ξA ≃ξX, the measured values of RX andˆRA would give mu/md ≃0.3. However we already argued that there is no reasonfor |B(φ)| ≪|A(φ)|.

Furthermore once we include B(φ) in the analysis as it mustbe, instantons give a potentially large positive contribution ωmax to ξA, while ξXreceives a negative contribution with the same size. This is essentially due to theinstanton-induced O(M2) piece in B(π0), i.e.

the term with the coefficient β arisingfrom the instanton-induced mass renormalization. Then what we obtain from theentire analysis can be summarized by the equation:md −mumd + mu≃0.54 (1 + ms4πf (ξA −ξX)),(27)where δξ ≡(ξA −ξX) is a totally unknown coefficient of O(1).

If ωmax = 2 ∼3 whosepossibility was assured within the semiclassical instanton gas approximation [1, 3], itis conceivable to assume that δξ is dominated by the positive instanton contribution2ωmax, implying ξA > ξX.Then the measured values of RX and ˆRA would givemu/md < 0.3. In any case, in order for mu/md to be precisely determined, we stillneed to compute (within QCD) the chiral coefficient δξ which is very sensitive to theITTo conclude, we have examined whether the QCD multipole expansion appliedfor the quarkonium decays Φ′ →Φ φ can be useful for the precise determinationof mu/md.The matrix element B(φ) which has been ignored in previous works13

can significantly affect the estimate of mu/md, particularly through the O(M2)-pieceinduced by instantons. As a result, a rather wide range of mu/md (including zero)can be consistent with the observed quarkonium decays.

As was concluded in ref. [7],to determine mu/md precisely, we need a QCD calculation of the chiral coefficientδξ = (ξA −ξX) which is sensitive to the IT.

If instanton contribution to δξ dominatesover other contribution, which is conceivable for ωmax = 2 ∼3, and thus δξ > 0,the observed quarkonium decays imply mu/md < 0.3, allowing the massless up quarkscenario [1, 2, 3] for the absence of CP violation in strong interactions.AcknowledgementsI thank A. Manohar for bringing my attention to ref. [6] and also for discussions.This work was supported by the DOE contract #DE-FG03-90ER40546.14

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