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RAL-92-026, CEBAF-TH-92-13, IU/NTC-92-16

arXiv:hep-ph/9301253v1 21 Jan 1993RAL-92-026, CEBAF-TH-92-13, IU/NTC-92-16Scalar Mesons in φ Radiative Decay:their implications for spectroscopy and for studies of CP-violation at φ factoriesF. E. CloseRutherford Appleton LaboratoryChilton, Didcot, Oxon, OX11 0QX, England.Nathan IsgurCEBAF12000 Jefferson Avenue, Newport News, VA 23606, U.S.A.S.

KumanoNuclear Theory CenterIndiana University, 2401 Milo B. Sampson Lane, Bloomington, IN 47408, U.S.A.AbstractExisting predictions for the branching ratio for φ →K ¯Kγ via φ →Sγ (whereS denotes one of the scalar mesons f0(975) and a0(980)) vary by several orders ofmagnitude. Given the importance of these processes for both hadron spectroscopyand CP-violation studies at φ factories (where φ →K0 ¯K0γ poses a possiblebackground problem), this state of affairs is very undesirable.We show thatthe variety of predictions is due in part to errors and in part to differences inmodelling.

The latter variation leads us to argue that the radiative decays ofthese scalar states are interesting in their own right and may offer unique insightsinto the nature of the scalar mesons. As a byproduct we find that the branchingratio for φ →K0 ¯K0γ is <∼0(10−7) and will pose no significant background toproposed studies of CP-violation.1

1IntroductionThere are predictions in the existing literature for the branching ratio forφ →K ¯Kγ via φ →Sγ (where S denotes one of the scalar mesons S∗(now calledf0(975)) or δ (now called a0(980)) that vary by several orders of magnitude [1-5]. Clearly not all of these predictions can be correct!

Given the importance ofthese processes for both hadron spectroscopy and CP-violation studies, this stateof affairs is clearly undesirable. Moreover, in view of the impending φ factory,DAΦNE [6], and other developing programmes [7], there is an urgent need toclarify the theoretical situation.The scalar mesons (i.e., mesons with JP Cn = 0++) have been a persistentproblem in hadron spectroscopy.∗We shall show in this paper that the radiativedecays of the φ meson to these states can discriminate among various models oftheir structure.In addition to the spectroscopic issues surrounding the scalarmesons, there is a significant concern that the decay φ →K0 ¯K0γ poses a possiblebackground problem to tests of CP-violation at future φ factories: the radiatedphoton forces the K0 ¯K0 system to be in a C-even state, as opposed to the C-odd decay φ →K0 ¯K0.

Looking for CP-violating decays in φ →K0 ¯K0 has beenproposed as a good way to measure ε′/ε [10], but because this means looking fora small effect, any appreciable rate for φ →K0 ¯K0γ (namely, a branching ratioφ →K0 ¯K0γ >∼10−6) will limit the precision of such an experiment. Estimates[4] of the non-resonant φ →K0 ¯K0γ rate give, in the absence of any resonantcontribution, a branching ratio of the order of 10−9, far too small to pose a prob-lem.

The uncertainty in the theoretical estimates, and the potential experimentalramifications, arise due to the presence of the scalar mesons f0(975) and a0(980),which are strongly coupled to the K ¯K system. Estimated rates for the resonantdecay chain φ →S + γ, followed by the decay S →K0 ¯K0, vary by three orders ofmagnitude, from a branching ratio of the order of 10−6 down to 10−9.

These vari-ations in fact reflect the uncertainties in the literature for the expected branchingratio for φ →Sγ which vary from 10−3 to 10−6 [11] . Here we concentrate on thisresonant process.We shall show that the variability of the predictions for φ →Sγ is due inpart to errors and in part to differences in modelling.

On the basis of this modeldependence, we argue that the study of these scalar states in φ →Sγ may offerunique insights into the nature of the scalar mesons. These insights should help∗For an historical perspective see Ref.

[8]; for a more recent study see Ref. [9].2

lead in the future to a better understanding of not only quarkonium but alsoglueball spectroscopy. As a byproduct we predict that the branching ratio forφ →K0 ¯K0γ is <∼0(10−7) (i.e., the branching ratio for φ →Sγ is <∼0(10−4))and will pose no significant background to studies of CP-violation at DAΦNE.2Probing the nature of the scalar mesons below 1 GeVThe scalar mesons are spectroscopically interesting for several reasons.

Oneis that, while agreeing on little else, it is an essentially universal prediction oftheory (lattices, bags, flux tube models, QCD sum rules, . .

. ) that the lowest-lying glueball has scalar quantum numbers and a mass in the 1.0 - 1.5 GeV massrange.

Clarifying the presently confused nature of the known 0++ mesons may bepivotal in the quest to identify this glueball. Another is the possibility that thetwo best known [12] scalar mesons, the f0(975) and the a0(980), are qq¯q¯q states.The original proposal [13] for this interpretation, based on the bag model, alsopredicted many other states which have not been seen (although this shortcomingis now understood to some degree [14]).

The qq¯q¯q interpretation of these twostates was later revived in a different guise within the quark potential model asthe “K ¯K molecule” interpretation [15]. Since providing a test of this particularinterpretation is one of the main results to be presented here, we first brieflyelaborate on these two models of multiquark states.In the naive bag model the qq¯q¯q states consist of four quarks confined in asingle spherical bag interacting via one gluon exchange.

It is obvious that such aconstruction will lead to a rich spectroscopy of states. Although it is not clear howto treat or interpret the problem of the stability of this spectrum under fission intotwo bags [14], it is very interesting that the dynamics of this model predicts thatthe lowest-lying such states will (in the SU(3) limit) form an apparently ordinary(“cryptoexotic”) nonet of scalar mesons.

It is, moreover, probable that a betterunderstanding of bag stability could solve both the problem of unwanted extrapredicted states and also a problem with the a0 itself: in the naive model it can“fall apart” into πη so that it is difficult to understand its narrow width, giventhe presently accepted pseudoscalar meson mixing angle (see footnote 22 in thefirst of Refs. [13]).

In the absence of an understanding of how to overcome thesedifficulties, we will not consider the bag picture further in this paper†.†See, however, Refs. [16] for a possible way out of the a0 →πη problem.3

In the potential model treatment [15] it is found that the low-lying qq¯q¯q sectoris most conveniently viewed as consisting of weakly interacting ordinary mesons:the resulting spectrum is normally a (distorted) two particle continuum. Withinthe ground state u,d,s meson-meson systems, the one plausible exception to thisrule is found in the K ¯K sector (i.e., the K ¯K channel and those other channelsstrongly coupled to it): the L = 0 (i.e., JP Cn = 0++) spectrum seems to havesufficient attraction to produce weakly bound states in both I = 0 and I = 1.There are a number of phenomenological advantages to the identification of thesetwo states with the f0(975) and a0(980).

Among them are:1) It is immediately obvious why the f0(975) and a0(980) are found just belowK ¯K threshold: they bear much the same relationship to it that the deuteron bearsto np threshold.2) The problem of the f0 and a0 widths is solved. If these states were 3P0quarkonia with flavours corresponding to ω and ρ (as suggested by their degener-acy), then Γ(f0 →ππ)/Γ(a0 →πη) would be about 4 in contrast to the observedvalue of about 12.

At least as serious is the problem in the quarkonium picturewith the absolute widths of these states: models [17-19] predict, for example,Γ(f0 →ππ)≃(3 −6)Γ(b1 →(ωπ)S)(2.1)≃500 −1000MeV(2.2)versus the observed partial width of 25 MeV. We have already noted the problem inthe bag model qq¯q¯q interpretation with a0 →πη.

In the K ¯K molecule picture thenarrow observed widths are a natural consequence of weak binding: (K ¯K)I=0 →ππ and (K ¯K)I=1 →πη occur slowly because the K ¯K wavefunction is diffuse.3) Both the f0 and a0 seem to bear a special relationship to s¯s pairs: theirK ¯K “couplings” are very large and they are observed in channels which wouldviolate the Okubo-Zweig-Iizuka (OZI) rule [20] for an ω, ρ -like pair of states [21].4) The γγ couplings of the f0 and a0 are about an order of magnitude smallerthan expected for 3P0 quarkonia [22], but consistent with the expectations for K ¯Kmolecules [23].Although these observations argue against the viability of the 3P0 quarkoniuminterpretation of the f0(975) (and probably also the a0(980)), they are insufficientto rule it out completely. (Moreover, a unitarized variant of the quark model [24],in which the scalar mesons are strongly mixed with the meson-meson continuum,avoids several of these problems.

In addition to this conservative alternative, therecent analysis of Ref. [9] has raised the possibility that the f0(975) is really acombination of two effects, one of which is a candidate for a scalar glueball.

)4

The main purpose of this paper is to point out a simple (and to us unexpected)experimental test which sharply distinguishes among these alternative explana-tions. We show that the rates for φ →f0(975)γ →ππγ and φ →a0(980)γ →πηγin the quarkonium, glueball, and K ¯K molecule interpretations differ significantly;furthermore, the ratio of branching ratiosφ →a0(980)γφ →f0(975)γalso may prove to be an important datum in that it can have a model-dependentvalue anywhere from zero to infinity (see Table 2)!In the quarkonium interpretation, φ →f0(975)γ and φ →a0(980)γ are sim-ple electric dipole transitions quite similar in character to several other mea-sured electric multipole transitions, including not only the light quark transitionsa2(1320) →πγ, K∗(1420) →Kγ, a1(1275) →πγ, and b1(1235) →πγ, but alsosuch decays as χc0 →ψγ and χb0 →Υγ.

From the comparison between theoryand experiment given in Ref. [17], we expect that the quark model predictionsfor these processes given in Table 1 are reliable to within a factor of two.

Thus ifthe f0 is an s¯s quarkonium, the branching ratio for φ →Sγ would typically be ofthe order of 10−5.If the f0(975) is a glueball (in Ref. [9] there is a glueball component of the“S∗effect”, dubbed the S1(991), which couples to ππ and is responsible for theresonant behaviour seen in ππ phase shift analyses; the other component, dubbedthe S2(998), is practically uncoupled to ππ) then one would naturally expectφ →f0(975)γ →ππγ to be even smaller than in the quarkonium interpretationsince the decay would be OZI-violating.

The remarks made above on the strongdecay widths of the quarkonium states would suggest that quarkonium - glueballmixing, through which we presume the OZI-violation would proceed, must besmall for the f0 (975) to remain narrow. Thus we can crudely estimate the glueball- quarkonium mixing angle to be less than [Γ(f0 →ππ)/Γ(3P0 →ππ)]12 so that iff0 (975) is a glueballΓ(φ →f0 (glueball) γ)≤Γ(f0 →ππ)Γ(3P0 →ππ)Γ(φ →f0 (quarkonium) γ) (2.3)≤120Γ(φ →f0 (quarkonium) γ)(2.4)Thus if f0 (975) is a glueball, this branching ratio should be more than an orderof magnitude smaller than it would be to a φ-like quarkonium.5

If the f0 is a quarkonium consisting only of nonstrange flavours, with a0 itsisovector quarkonium partner, these states will be OZI decoupled in the φ radiativedecay. The OZI-violating production rate via a K ¯K loop, viz.

φ →γK ¯K →γa0,may be calculated. This calculation reveals some interesting points of principlewhich shed light on the role of finite hadron size in such loop calculations; thiscalculation will be discussed in the next section.Interesting questions arise in the case of qq¯q¯q or K ¯K bound states (“molecules”).The quark contents of these two systems are identical but their dynamical struc-tures differ radically.

The situation here has its analog in the case of the deuteronwhich contains six quarks but is not a “true” six-quark bound state. The essen-tial feature is whether the multiquark system is confined within a hadronic systemwith a radius of order (ΛQCD)−1 or is two identifiable colour singlets spread over aregion significantly greater than this (with radius of order (µE)12 associated withthe interhadron binding energy E for a system of reduced mass µ).

In the formercase the branching ratio may be as large as 10−4 (see Ref. [5] and section 4);the branching ratio for a diffuse K ¯K molecular system can be much smaller, asdiscussed below.The ratio of branching ratios is also interesting.

The ratio of Γ(φ →γa0)/Γ(φ →γf0) is approximately zero if they are quarkonia (the f0 being s¯s and the a0 beingOZI decoupled), it is approximately unity if they are K ¯K systems, while for q2¯q2the ratio is sensitively dependent on the internal structure of the states. Thissensitivity in qq¯q¯q arises because φ →Sγ is an E1 transition whose matrix ele-ment, being proportional to Σei⃗ri, probes the electric charges of the constituentsweighted by their vector distance from the overall centre of mass of the system.Thus, although the absolute transition rate for S = qq¯q¯q depends on unknowndynamics, the ratio of a0 to f0 production will be sensitive to the internal spa-tial structure of the scalar mesons through the relative phases in I = 0 and 1wavefunctions and the relative spatial distributions of quarks and antiquarks.For example, suppose that the state’s constituents are distributed about thecentre of mass with the structure (q¯s)(¯qs), where q denotes u or d, and (ab)represents a spherically symmetric cluster.

Thenf0a0=1√2[(u¯s)(¯us) ± (d¯s)( ¯ds)](2.5)and the E1 matrix element will beM ∼[(eu + e¯s) ± (ed + e¯s)] = eK+ ± eK06

and hence the ratio Γ(φ →γf0)/Γ(φ →γa0) will be unity.The quarks aredistributed as if in a K ¯K molecular system (which is a specific example of thisconfiguration) and only the absolute branching ratio will distinguish q2¯q2 fromK ¯K.If the distribution is (q¯q)(s¯s) then the matrix elementM ∼[(eq + e¯q) −(es + e¯s)] = 0.Here the quark distributions mimic π0η (in the a0) or ηη (in the f0). In thiscase the absolute branching ratios will be suppressed.

Most interesting is the casewhere S = D ¯D, where D denotes a diquark, i.e. wheref0a0=1√2[(us)(¯u¯s) ± (ds)( ¯d¯s)](2.6)in which caseM ∼[(eu + es) ± (ed + es)]so thatΓ(φ →γa0)Γ(φ →γf0) = (1 + 21 −2)2 = 9.The absolute rate in this case depends on an unknown overlap between K ¯K andthe diquark structure; nonetheless the dominance of a0 over f0 would be ratherdistinctive.

For convenience these possibilities are summarised in Table 2.3The K ¯K Loop Contribution to φ →SγThe φ and the S (where S = a0 or f0) each couple strongly to K ¯K, with thecouplings gφ and g for φK+K−and SK+K−being related to the widths byΓ(φ →K+K−) =g2φ48πm2φ(m2φ −4m2K+)3/2(3.1)andΓ(S →K+K−) =g216πm2S(m2S −4m2K+)1/2(3.2)for kinematical conditions where the decay is allowed.Hence, independent ofthe dynamical nature of the S, there is an amplitude M(φ →Sγ) for the decayφ →Sγ to proceed through the charged K loop (fig. 1), φ →K+K−→S(ℓ) + γ7

where the K± are real or virtual and S is the scalar meson with four momentumℓ. The amplitude describing the decay can be writtenM(φ(p, ǫφ) →S(ℓ)+γ(q, ǫγ)) =egφg2π2im2KI(a, b)[(p·q)(ǫγ ·ǫφ)−(p·ǫγ)(q·ǫφ)] (3.3)where ǫγ and ǫφ (q and p) denote γ and φ polarisations (momenta).The quantities a, b are defined as a =m2φm2K , b =ℓ2m2K so that a −b =2p.qm2K isproportional to the photon energy, and I(a, b) which arises from the loop integralisI(a, b) =12(a −b) −2(a −b)2{f(1b) −f(1a)} +a(a −b)2{g(1b) −g(1a)}(3.4)wheref(x)=−(arcsin(12√x))2x > 1414[ln( η+η−) −iπ]2x < 14g(x)=(4x −1)1/2 arcsin(12√x)x > 1412(1 −4x)1/2[ln( η+η−) −iπ]x < 14η±=12x(1 ± (1 −4x)1/2)(3.5)Note that ℓ2 may in general be virtual, though we shall here concentrate on thereal resonance production where ℓ2 = m2S with mS ≃975 or 980 MeV.Even though Refs.

[1-4] use essentially the same values for the couplings andother parameters, they obtain different results. Our results confirm those of Ref.

[1] apart from a minor numerical error. Ref.

[5] claims that the value of the loopcalculation depends on the dynamical nature of the S. Since the coupling S →K ¯Kis input from data it is somewhat surprising that the result can discriminateamongst models of the S. We confirm the numerical result of Ref. [5] and discussits physical significance below.The resonant contributions to the φ →K0 ¯K0γ branching fraction give a dif-ferential decay widthdΓdk2 = |I(a, b)|2g2ϕg24m4Kπ4χ(3.6)where χ is given byχ =α128π2m3ϕ13(m2ϕ −ℓ2)3(1 −4m2Kℓ2 )1/2(ℓ2 −m2S)2 + m2SΓ2S(3.7)8

Here ℓ2 is the invariant mass squared of the final K0 ¯K0 system, and hence theresonance.The limitations and problems in the existing literature concerning attemptsto calculate the above are discussed in Ref. [11].

Here we shall briefly reviewthe loop calculation in order to assess the existing literature and to highlight thenovel features of the case where the S is a K ¯K bound state with a finite size.Calculation of the integral I(a, b)Upon making the φ and K interactions gauge invariant, one finds for chargedkaonsHint = (eAµ + gφφµ)jµ −2egφAµφµK†K(3.8)where Aµ, φµ and K are the photon, phi and charged kaon fields, jµ = iK†(⃗∂µ −←∂µ)K. If the coupling of the kaons to the scalar meson is assumed to be simplythe point-like one SK+K−, then gauge invariance generates no extra diagram andthe resulting diagrams are in figs. (1).

Immediately one notes a problem: thecontact diagram fig. (1a) diverges.

The trick has been to calculate the finite sumof figs. (1b) and (1c) and then, by appealing to gauge invariance, to extract thecorrect finite part of fig.

(1a). This is done either bya) (Refs.

[1-3]) Fig. (1a) contributes to Aνφµgµν whereas figs.

(1b) and (1c)contribute both to this and to pνqµAνφµ. Therefore one need calculate only thelatter diagrams, since the finite coefficient of the pνqµ term determines the resultby gauge invariance.b) (Ref.

[5]) These authors compute the imaginary part of the amplitude(which arises only from figs. (1b) and (1c)) and write a subtracted dispersion rela-tion, with the subtraction constrained by gauge invariance.

This is also sufficientto determine the amplitude.In section 4 we shall consider the case where the scalar meson is an extendedobject, in particular a K ¯K bound state. The SK ¯K vertex in this case involves amomentum-dependent form factor f(k), where k is the kaon, or loop, momentumwhich will be scaled in f(k) by k0, the mean momentum in the bound statewavefunction or, in effect, the inverse size of the system.In the limit whereR →0 (or k0 →∞) we recover the formal results of approaches (a, b) above, aswe must, but our approach offers new insight into the physical processes at work.In particular, in this more physical case there is a further diagram (fig.

(2d))9

proportional to f ′(k) since minimal substitution yieldsf(|⃗k −e ⃗A|) −f(|⃗k|) = −e ⃗A · ˆk∂f∂k(3.9)As we shall see, this exactly cancels the contribution from the seagull diagram fig. (2a) in the limit where qγ →0, and gives an expression for the finite amplitudewhich is explicitly in the form of a difference M(q) −M(q = 0).

This makescontact with the subtracted dispersion relation approach of Ref. [5].First let us briefly summarise the calculation of the Feynman amplitude in thestandard point-like field theory approach, as it has caused some problems in Refs.[2,3].

If we denote Mµν = [pνqµ −(p.q)gµν]H(mφ, mS, q) (see eq. (3.3)), then thetensor for fig.

(3) may be written (compare with eqs. 8 and 6 of Refs.

[2] and [3],respectively)Mµν = eggφZd4k(2π)4(2k −p)µ(2k −q)ν(k2 −m2K)[(k −q)2 −m2K][(k −p)2 −m2K](3.10)We will read offthe coefficient of pνqµ after combining the denominators by thestandard Feynman trick so thatMµν = eggφ(2π)48Z 10 dzZ 1−z0dyZ ∞−∞d4kkµkν[(k −qy −pz)2 −c + iǫ]3(3.11)where c ≡m2K −z(1 −z)m2φ −zy(m2S −m2φ). The pνqµ term appears when wemake the shift k →k + qy + pz to obtainH = eggφ4π2iZ 10 dzZ 1−z0dy yz[m2K −z(1 −z)m2φ −zy(m2S −m2φ)]−1.

(3.12)Note that m2S < m2φ and so one has to take care when performing the y integration.One obtains (recall a = m2φ/m2K, b = m2S/m2K)H =eggφ4π2im2K1(a −b){Z 10dzz [z(1 −z) −(1 −z(1 −z)a)(a −b)ln( 1 −z(1 −z)b1 −z(1 −z)a)]−iπ(a −b)Z 1/η−1/η+(1 −z(1 −z)a)dzz }(3.13)where η± ≡a2(1 ± ρ) with ρ ≡q1 −4/a. (In performing the integrals, one musttake care to note that a > 4 whereas b < 4 (which causes ρ2a > 0, ρ2b < 0)).

Ourcalculation has so far only taken into account the diagram where the K+ emitsthe γ; the contribution for the K−is identical, so the total amplitude is doublethat of eq. (3.13) and therefore in quantitative agreement with eqs.

(3) and (4)of Ref. [1].

Straightforward algebra confirms that this agrees with eqs. (9-11) ofRef.

[5].10

Numerical evaluation, using m(f0) = 975 MeV and g2/4π = 0.6 GeV2 leads toΓ(φ →f0γ) = 6 × 10−4MeV(3.14)somewhat at variance with the value of 8.5 × 10−4 MeV quoted in Ref. [1] ‡.Ref.

[5] does not directly quote a rate for φ →f0γ. Instead, it quotes values forφ →γf0 →γππ (for example) and claim that this depends upon the q¯q or q2¯q2structure of the f0.

However, the differences in rate (which vary by an order ofmagnitude between q¯q and q2¯q2 models) arise because different magnitudes for thefK ¯K couplings have been used in the two cases. In the q2¯q2 model a value forg2(fK ¯K) was used identical to ours and, if one assumes a unit branching ratiofor f0 →ππ, the rate is consistent with our eq.

(3.14) (Ref. [5] has integratedover the resonance).

In the case of the a0, Ref. [5] notes that in the q2¯q2 modelthe relation between g2(a0K ¯K) and g2(a0πη) implies Γ(a0 →πη) ≃275 MeV.In the q¯q model, in contrast, Ref.

[5] uses as input the experimental value ofΓ(a0 →πη) ≃55 MeV which implies a reduced value for g2(a0πη) and, therefore,for g2(a0 ¯KK): the predicted rate for φ →γa0 →γπη is correspondingly reduced.Thus we believe that the apparent structure-dependence of the reaction φ →Sγ claimed in Ref. [5] is suspect.

The calculation has assumed a point-like scalarfield which couples to point-like kaons with a strength that can be extracted fromexperiment. The computation of a rate for φ →K ¯K →γS will depend upon thisstrength and cannot of itself discriminate among models for the internal structureof the S.We shall now consider the production of an extended scalar meson [11] whichis treated as a K ¯K system (based on the picture developed in Refs.

[15]).4K ¯K loop production of an extended scalar mesonSuppose that K+ and K−with three momenta ±⃗k produce an extended scalarmeson in its rest frame. The interaction Hamiltonian H = gφ(|⃗k|)SK+K−is ingeneral a function of momentum.

Now make the replacement ⃗k →⃗k −e ⃗A, andexpand φ(|⃗k −e ⃗A|) to leading order in e; one then finds a new electromagneticcontributionHK+K−f0γ = −egφ′(k)ˆk · ⃗A . (4.1)‡ However, J. Pestieau, private communication, confirms our value.11

The finite range of the interaction, which is controlled by φ(k), implies that thecurrents flow over a finite distance during the K ¯K →S transition: this current isthe “interaction current”. The above current given by minimal substitution is notunique, in the sense that the transverse part ⃗ǫγ · ⃗j cannot be determined by therequirement of gauge invariance alone.

However, it should describe the processunder consideration accurately since the radiated photon is soft: the details ofthe interaction current are not important in the soft photon regime [25]. Theeffect of this form factor is readily seen in time ordered perturbation theory.

(Inthis section we will work in the non-relativistic approximation. This suffices bothto make our point of principle and to provide numerically accurate estimates fornonrelativistic K ¯K bound states such as the f0 and a0 in the Ref.

[15] picture. Ingeneral there are further time orderings whose sum gives the relativistic theory;see below.

)There are four contributions: (H1,4 correspond to figs. (2a) and (2d), whileH2,3 correspond to figs.

(2b) and (2c), where the γ is emitted from the K+ or K−leg). We write these (for momentum routing see fig.

(3))H2,3=2eggφZd3kφ(k)2⃗ǫγ.⃗k(⃗k.⃗ǫφ ± 12⃗q.⃗ǫφ)D(E)D1D(q±)(4.2)H1=2eggφZd3kφ(k)⃗ǫγ.⃗ǫφD1(4.3)H4=2eggφZd3kφ′(k)⃗ǫγ.ˆk⃗ǫφ.⃗kD(0)(4.4)whereD1≡mφ −q −D(E)D(q±)≡mφ −2E±(4.5)D(0)≡mφ −2E(k)(4.6)D(E)≡E+ + E−(4.7)and where E± = E(k ± q/2) with E(P) the energy of a kaon with momentumP. Note that H1 is the (form-factor-modified) contact diagram and H4 is the newcontribution arising from the extended SK ¯K vertex.After some manipulations their sum can be writtenH = 2eggφ⃗ǫγ.⃗ǫφZd3k[φ(k)D1{1+⃗k2 −(⃗k.ˆq)2D(E)(1D(q+) +1D(q−))}+ φ′(k)|⃗k|3D(0) ] .

(4.8)12

If limk2→∞(k2φ(k)) →0 § we may integrate the final term in eq. (4.8) by partsand obtain for itH4 = 2eggφ⃗ǫγ.⃗ǫφZd3k φ(k)D(0){−1 −⃗k2 −(⃗k.ˆq)2E(k)D(0) }(4.9)This is identical to the ⃗q →0 limit of H1+H2+H3, and hence we see explicitly thatthe gµν term (i.e., the term proportional to ⃗ǫγ.⃗ǫφ as calculated above) is effectivelysubtracted at ⃗q = 0 due to the partial integration of the φ′(k) contribution, H4.If one has a model for φ(k) one can perform the integrals numerically.

For theK ¯K molecule, the wavefunctionψ(r) =1√4πu(r)r(4.10)is a solution of the Schrodinger equation{−1mKd2dr2 + v(r)}u(r) = Eu(r). (4.11)One may approximate (see Ref.

[23])v(r) = −440(MeV ) exp(−12( rr0)2)(4.12)with r0 = 0.57 fm.Equation (4.11) may be solved numerically, giving E =−10 MeV and a ψ(r) which for analytic purposes may, as we shall see, be well-approximated byψ(r) = (µ3π )1/2 exp(−µr);µ ≡√32RK ¯K(4.13)where RK ¯K ≃1.2fm (thus ψ(0) = 3 × 10−2GeV 3/2; see also Ref. [23]).

Themomentum space wave function that is used in our computation (see fig. (4)) isthus taken to haveφ(k)φ(0) =µ4(k2 + µ2)2.

(4.14)The rate for Γ(φ →Sγ) is shown as a function of RK ¯K in fig. (5).

The nonrela-tivistic approximation eqs. (4.2-4.9) is valid for RK ¯K >∼0.3fm which is applicableto the K ¯K molecule: for RK ¯K →0 the fully relativistic formalism is required andhas been included in the curve displayed in fig.

5. As RK ¯K →0 and φ(k) →1 werecover the numerical result of the point-like field theory, whereas for the specific§Actually, when k →∞the relativistic expressions of the next subsection are needed.

Theseshow that φ(k) need only vanish logarithmically to obtain convergence.13

K ¯K molecule wavefunction above one predicts a branching ratio of some 4×10−5(width ≃10−4MeV ). This is only 15 of the point-like field theory result but islarger than that expected for the production rate of an s¯s scalar meson (see Tables1 and 2).Connection with Relativistic Field TheoryThe nonrelativistic formalism is sufficient for describing the radiation from aK ¯K molecule.

However, it does not have the proper limit as RK ¯K →0; in thislimit relativistic K ¯K pairs are important in the loop integral. In this section weshow how the relativistic formalism can be obtained from time-ordered perturba-tion theory and make contact with the relativistic field theory formalism of section3.

The matrix elements for the time-orderings of fig. (6) areMµ1=+ ieggφZd3k(2π)3 φ(|⃗k|) 2εµφ [ −12E+2E−(ES + E+ + E−)+12E+2E−(mφ −q −E+ −E−) ](4.15)where the first (second) term corresponds to fig.

(6a) (fig. (6b)) and E± is definedby E± = E(k ± q/2).

Using ES = mφ −q,−1mφ −q + E+ + E−= +2E+(mφ −q + E−)2 −E2+−1mφ −q + E−−E+(4.16)and1mφ −q −E+ −E−= +2E−(mφ −q −E+)2 −E2−+1mφ −q −E+ + E−,(4.17)we obtainMµ1=+ ieggφZd3k(2π)3 φ(|⃗k|) 2εµφ [12E−[(mφ −q + E−)2 −E2+]+12E+[(mφ −q −E+)2 −E2−] ]. (4.18)Analogously, Mµ2 , Mµ3 , and Mµ4 areMµ2= Mµ3= + ieggφZd3k(2π)3 φ(|⃗k|) 2εµφ [⃗k2 −(⃗k · ˆq)2]× [12E+[(q + E+)2 −E2−][(mφ + E+)2 −E2+]14

+12E−[(q −E−)2 −E2+][(mφ −q + E−)2 −E2+]+12E+[(mφ −E+)2 −E2+][(mφ −q −E+)2 −E2−] ](4.19)Mµ4=+ ieggφZd3k(2π)3 φ′(|⃗k|) |⃗k|32εµφ [12E0[(mφ + E0)2 −E20]+12E0[(mφ −E0)2 −E20] ](4.20)where E0 is defined by E0 = E(k).In this way, we obtain “relativistic” expressions for the radiative φ mesondecays. Matrix elements for the process a−d in fig.

(2) may thus be writtenMµ1=−eggφZd4k(2π)4 φ(|⃗k|)2εµφD(k −q/2)D(k + q/2 −p)(4.21)Mµ2=+ eggφZd4k(2π)4 φ(|⃗k|)εφ · (2k + q −p) (2k)µD(k + q/2)D(k −q/2)D(k + q/2 −p)(4.22)Mµ3=+ eggφZd4k(2π)4 φ(|⃗k|)εφ · (2k −q + p) (2k)µD(k + q/2)D(k −q/2)D(k −q/2 + p)(4.23)Mµ4=+ eggφZd4k(2π)4 φ′(|⃗k|) εφ · (2k −p) ˆkµD(k)D(k −p)(4.24)where D(k) is defined byD(k) = k2 −m2K(4.25)and ˆk = (0,⃗k/|⃗k|). In the particular case where φ(|⃗k|) = 1 and φ′(|⃗k|) = 0, thesereproduce the familiar field theory expressions of Refs.

[1-5] and section 3. It isinteresting to note the role that φ′(|⃗k|) plays in regularising the infinite integral.Define the matrix elements ˜Mj (j = 1 −4) by ˜Mj = εγ · Mj/[ieεγ · εφ] and thedecay width is then calculated byΓ(φ →Sγ) =αq3m2φ| ˜M|2,˜M = ˜M1 + ˜M2 + ˜M3 + ˜M4(4.26)which reproduces the expressions in Refs.

[1-5] and provides a check on ourformalism. Eqs.

(4.21-4.24), when evaluated numerically, give the decay widthsshown in Fig. 5.

In the limit RK ¯K →0 our numerical results agree with eq. (3.14)which was obtained by using the point-like field theory.15

5A Comment on the OZI RuleThe calculations presented in this paper may have a bearing on one of theleast understood characteristics of the low energy strong interactions: the Okubo-Zweig-Iizuka (OZI) rule [20]. If the a0 were a1√2(u¯u −d ¯d) state, its productionin φ →a0γ would vanish in “lowest order” in the quark model, with the K ¯Kloop contribution presumed to provide a small correction since such processes areOZI-violating (e.g., ω −φ mixing could also occur via such loops).

We have seenthat in the point-like approximation φ →a0γ would proceed with a branchingratio of order 10−4 via this loop process, as would f0 =1√2(u¯u + d ¯d). If f0 = s¯s,a similar rate would be obtained from the K ¯K loop, but now there would be adirect term which is supposed to be dominant.

It is, however, easy to discoverthat this direct process would only produce a branching ratio of the order of 10−5(see Table 1).Our calculation provides some insight into this conundrum. If the K ¯K systemis diffuse, RK ¯K >∼2fm, then the loop calculation gives a branching ratio < 10−5(see fig.

(5)) and the empirical OZI rule is good. Physically, the rate is suppresseddue to the poor spatial overlap between the K ¯K system and the φ.

The point-likefield theory does not allow for this: superficially the loops have a large magnitude.The essential observation is that the point-like calculation does not take intoaccount the confinement scale, even though it is clear from our results that thedynamics can depend on it rather critically.Now consider a φ and assume that S is an (s¯s) scalar meson, confined inΛ−1QCD ≃1 fm and connected by an intermediate state with quark compositionq¯qs¯s. If this multiquark system is confined in a length scale <∼Λ−1QCD ≃1fm(i.e., it is a “genuine” q2¯q2 state and separate identifiable kaons are not present),then the point-like field theory calculations, which contain no intrinsic length, aresuperficially at least roughly applicable.

The φ →γS branching ratio via the K ¯Kpart of this compact system is then elevated above the 10−5 barrier. However, if apure K ¯K intermediate state forms, then it must occupy > 2Λ−1QCD.

The amplitudefor the φ or a S(s¯s) to fluctuate to this scale of size would be small and it is thissupression that is at the root of the OZI rule in this process.We see from this reasoning that the contribution of diagrams which correspondat the quark level to q¯qs¯s loops really contain two distinct contributions at thehadronic level. These are first of all the diffuse contributions which can arise fromhadronic loops corresponding to nearby thresholds, in this case from K ¯K.

Then16

there are “short distance” contributions where approximating the q¯qs¯s systemas a K ¯K system is potentially very misleading: a realistic calculation of suchcontributions would at least have to include a very large set of hadronic loops.A step in this direction has recently been taken in Refs. [26].

These authorshave considered the loop contributions to, e.g., ω −φ mixing in the 3P0 quarkpair creation model, and found that there is a systematic tendency for the sumof all hadronic loops to cancel.In fact, they show that (in their model) theincompleteness of the cancellation of OZI-violating hadronic loops is precisely dueto nearby thresholds.6ConclusionsThere is still much thought needed on the correct modelling of the K ¯K or q2¯q2scalar meson and the resulting rate for φ →Sγ: the present paper merely makesa start by clarifying the present literature, making the first predictions for theproduction of a K ¯K molecule, and pointing out the utility of the ratio of branchingratios as a filter. However, these results in turn raise questions that merit furtherstudy.For example, there are interesting interference effects possible betweenthe a0(I = 1) and f0(I = 0) states which have not been considered.

These twonearly degenerate states lie so near to the K ¯K thresholds that the mass differencebetween neutral and charged kaons is not negligible: for example, their widthsstraddle the K+K−threshold but only barely cross the K0 ¯K0 threshold (at leastin the case of the relatively narrow f0).Although there is clearly much to be done, it is already clear that there maybe unique opportunities for probing dynamics in φ →Sγ and investigating thenature of the scalar mesons below 1 GeV. Moreover, we can already conclude thatthe branching ratio of φ →Sγ will be between 10−4 and 10−5 depending on thedynamical nature of these scalars and so will generate nugatory¶ background tostudies of CP-violation at DAΦNE or other φ-factories.Acknowledgements¶ n¯ug’ atory, a. Trifling, worthless, futile; inoperative, not valid.

[f. L nugatorius (nuggaritrifle f. prec., -ORY)][28]17

We are indebted to the Institute for Nuclear Theory in Seattle for their hos-pitality. F.E.C.

would like to dedicate this paper to the memory of Nick Brown,who was very interested in this physics. He would also like to thank D. Rossand P. Valovisky for comments, the organisers of the DAΦNE workshops, and M.Pennington and G. Preparata for their interest in the OZI rule and some techni-cal aspects of this work.

N.I. would like to thank the Department of TheoreticalPhysics of the University of Oxford for its hospitality during the period whenthis work was begun and, with F.E.C., to express his gratitude to the refereewho rejected an earlier incorrect version of this paper [27].

S.K. acknowledgesthe support of the U.S. NSF under contract NSF-PHY91-08036; N.I.

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Figure CaptionsFigure 1. The contact (a) and loop radiation (b,c) contributions.Figure 2.

As fig. 1 but with an extended scalar meson.

Note the new diagram(d).Figure 3. Momentum routing.Figure 4.

Comparison between the exact momentum space wavefunction φ(k)(solid) and the approximation of eq. (4.14); k is the relative momentumof the K and ¯K.Figure 5.

Γ(φ →Sγ) in MeV versus RK ¯K in fm.Figure 6. The two time orderings of fig.

2(a).21

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