bound states and approximate Nambu-Goldstone bosons

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  • On the and K as q q bound states and approximate Nambu-Goldstone bosons

    Shmuel Nussinov1,2 and Robert Shrock3

    1School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel2Schmid College of Science, Chapman University, Orange, California 92866, USA

    3C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794, USA(Received 20 November 2008; published 23 January 2009)

    We reconsider the two different facets of and K mesons as q q bound states and approximate Nambu-

    Goldstone bosons. We address several topics, including masses, mass splittings between and and

    between K and K, meson wave functions, charge radii, and the K wave function overlap.DOI: 10.1103/PhysRevD.79.016005 PACS numbers: 11.30.Rd, 12.38.t, 12.39.Jh, 12.40.Yx

    I. INTRODUCTION

    Quantum chromodynamics (QCD) is most predictive inthe perturbative, short-distance regime. Yet our under-standing of long distance, nonperturbative properties ofQCD keeps improving. Lattice QCD has been used tocompute the hadronic spectrum and matrix elements forweak transitions, the Nc ! 1 limit and new variantsthereof have been widely applied, and heavy-quark sym-metry has helped to elucidate heavy-heavyQ Q and heavy-light Q q hadrons. The SU2L SU2R global chiralsymmetry and the spontaneous breaking of this symmetryto the vectorial, isospin SU2V , which are mandatory inQCD [113], underlie isospin symmetry, the fact that thepions are light almost Nambu-Goldstone bosons (NGBs)[14,15], and the usefulness of chiral perturbation theory.Another specific advance was the resolution of the U1Aproblem by instantons [16], thus explaining why the 0 isnot light.

    The remarkable success of the nonrelativistic (naive)quark model (NQM) treating the u, d, and s quarks asnonrelativistic, spin-1=2 fermions is of great interest [1719]. This model dictated the low-lying flavor SU(3) mul-tiplets and many aspects of their electroweak interactions.Spontaneous chiral symmetry breaking (SSB) generatesdynamical, constituent masses of order the QCD scaleQCD 250 MeV for the light, almost massless u and dquarks and increments, by this amount, the hardLagrangian mass of the s quark,ms 120 MeV to producea constituent s quark mass. Our results are not sensitivelydependent upon the values of the constituent quark masses;we use the values [20,21]

    Mu Md Mud 340 MeV (1.1)and

    Ms 470 MeV: (1.2)The dynamical mass generation of the constituent quarkmasses in QCD can, for example, be shown via analysis ofthe Dyson-Schwinger equation for the quark [22]. Onemay roughly characterize the range of the QCD interac-tions responsible for the h qqi condensate as r0. The effec-

    tive size of a constituent quark, consisting of the barevalence quark and its entourage of gluons and q q pairs,is expected to be of order r0. A small r0, less than 0.2 fm,say, then yields constituent quarks of that size movinginside a hadron of size approximately 1 fm under theinfluence of a smooth confining potential, making theNQM plausibly justified.Some of the mechanisms suggested for generating spon-

    taneous chiral symmetry breakingin particular, Nambu-Jona Lasino-type (NJL) models [14,23], and those involv-ing instantons [16,24,25]can have relatively short range.However, in the approach of Casher [1] and Banks andCasher [2], SSB results from confinement. In this ap-proach there is no separation of scales between the con-stituent quark size and hadron sizes.The almost massless Nambu-Goldstone pionthe other

    consequence of spontaneous chiral symmetry breakinggenerates arguably the single most serious difficulty for theNQM, namely, what has been called the - puzzle(e.g., [26]) This is the challenge of simultaneously explain-ing the pion as a q q bound state and an approximate NGB,and relating it to the . There is an analogous, although lesssevere, problem for the K and K. Although this difficultyis most clearly manifest within the NQM, it transcendsthis nonrelativistic model. Thus, also the originalMassachusetts Institute of Technology (MIT) bag modelwith relativistic quarks confined in a spherical cavity re-quires large hyperfine interactions to try to split the massesof the and and, like the NQM, fails to explain thealmost massless pions [27]. To get a sufficiently light pionin the MIT bag model, it is necessary to argue that sub-stantial contributions due to the fluctuations in the center-of-mass should be subtracted [28] (see also [29,30]).In this paper we shall revisit the problem of understand-

    ing the dual nature of the pion (and kaon) as q q boundstates and as collective, almost massless Nambu-Goldstonebosons. We suggest that this can be understood because thestrong chromomagnetic hyperfine interaction that splits themass of the and keeps the valence q and q in the pionrather close to each other. The dynamics of this close q qpair can be modeled by random walk methods, and weshow that these can reproduce the successful relation that

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    1550-7998=2009=79(1)=016005(15) 016005-1 2009 The American Physical Society

    http://dx.doi.org/10.1103/PhysRevD.79.016005

  • m2 / mq and can explain relations that have been chal-lenging to understand, such as the dependence of the pionwave function at the origin on the pion mass. The organi-zation of the paper is as follows. In Sec. II we elaborate onthe - puzzle in the context of the nonrelativistic quarkmodel. In Sec. III we present a heuristic picture that givessome new insight into this puzzle by helping to explain the as both a q q bound state and an approximate Nambu-Goldstone boson. In Sec. IV we consider the K tran-sition form factor, fq2. Its deviation from unity atvanishing momentum transfer is governed by theAdemollo-Gatto theorem [31], and analogous deviationsfrom heavy-quark universality are derived in a generalcontext. In Sec. V we comment on the systematics of quarkmass differences inferred from (Qs) and (Q q) mesons,where q u or d. This has been elaborated independentlyby Karliner and Lipkin [32]. Still, we feel that it is ofsufficient interest to present it here from our point of view.

    II. THE - PUZZLE

    In this section we briefly review the - puzzle. Thereader who is already familiar with this may wish to skimthis section or skip it and proceed immediately to Sec. IIIwhere we present our new physical picture of the pion.

    Several pieces of data suggest that the pion, which islighter than the by approximately 640 MeV, is otherwiserather similar to the , as expected in the nonrelativisticquark model for the 1S0 pseudoscalar partner of the

    3S1vector meson. These data can be summarized as follows:

    (i) One measure of the size of a hadron is provided bythe magnitude of the charge radius. The charge radiiof the and K are given by [20]

    hr2i1=2 0:672 0:008 fm (2.1)and

    hr2iK1=2 0:560 0:031 fm: (2.2)These are rather similar and, indeed, are not verydifferent from the charge radius of the proton,

    hr2ip1=2 0:875 0:007 fm: (2.3)

    (ii) Similar and sizes and a somewhat smaller kaonare suggested by the total cross sections on protonsat a typical laboratory energy above the resonanceregion. Averaged over meson charges, at a lab en-ergy Elab 10 GeV, these are [20]

    N 26:5 mb (2.4)and

    KN 21 mb: (2.5)Although one obviously does not have beams of mesons available experimentally, owing to the very

    short lifetime of the , it is possible to estimate whatthe cross section for N scattering would be ifone did have such beams. Diffractive productiondata and vector meson dominance yield the estimate[33]

    N 27 2 mb: (2.6)This cross section is essentially the same, to withinthe experimental and theoretical uncertainties, asN at the same energy [and these are approxi-mately equal to 2=3NN at this energy].

    (iii) The nonrelativistic quark model was able to fit themeasured values of the proton and neutron mag-netic moments p 2:793N and n 1:913N [where N e=2mp] and the ratiop=n 3=2, as well as the values of the hy-peron magnetic moments, in terms of Dirac mag-netic moments of constituent quarks. It alsoexplained decays such as ! ! 0 and ! as quark spin flip 3S1 ! 1S0 electromagnetictransitions. The optimal overlap of the and wave functions implied by this confirms the simi-larity of the vector and pseudoscalar meson ground-state wave functions.

    (iv) The amplitudes for semileptonic K3 decays in-volve the vector part of the weak jSj 1 currentand contain the product of Vus with the fq2transition form factor. In the limit of SU(3) flavorsymmetry mu md ms, so that mK m, theconserved vector current (CVC) property impliesthat f0 1. Experimentally, fq2 0 is veryclose to unity. The success of these fits impliesalmost optimal overlap between the wave functionsof the pion and kaon.

    In the nonrelativistic quark model, one can rewrite thetwo-body quark-antiquark Hamiltonian as an effectiveone-body problem with the usual reduced mass

    i j MiMj

    Mi Mj (2.7)

    for the qi qj pseudoscalar meson. (The context will make

    clear where the notation refers to a magnetic momentand where it refers to a reduced mass.) The correspondingbound-state wave function is denoted c r, c Kr, etc.,where r rqi r qj is the relative coordinate in the boundstate. With the above-mentioned typical values Mud 340 MeV and Ms 470 MeV, one has

    Mud2 170 MeV (2.8)and

    K MudMsMud Ms 200 MeV; (2.9)

    where it is understood that the choices for the input values

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  • of the constituent quark masses in these formulas dependsomewhat on the method that one uses to infer their values[21]. In the nonrelativistic quark model, since a bound stateinvolving a larger effective reduced mass is expected to besmaller, one has some understanding of the fact thatffiffiffiffiffiffiffiffiffiffiffiffiffihr2iKp 0:83 ffiffiffiffiffiffiffiffiffiffiffiffiffihr2i

    p. The deviation of f0 from unity

    is also in accord with this difference of reduced masses.For heavy Q Q quarkonium systems one can use the

    nonrelativistic Schrodinger equation to describe a numberof properties of the bound states [18,19]. This use isjustified by the fact that in the c c and b b systems therespective heavy-quark masses mc 1:3 GeV and mb 4:3 GeV are large compared with QCD, and the asymp-totic freedom of QCD means that s gets small for suchmass scales. The hyperfine splitting in these QQ systems,being proportional to s=mQ, is small.

    For light qq systems, however, the situation is different.Let us denote

    h0jJjjkpi ifjkp; (2.10)where j, k are isospin indices and J is the weak chargedcurrent, so that h0jJ1i2 jpi ifp. Similarly,h0jJ4i5 jKpi ifKp. Experimentally [20],

    f 92:4 MeV; fK 113 MeV: (2.11)Analogous constants enter in the leptonic decays of thevector mesons. The rate for the decay M

    i j! , where

    or e, is

    Mi j!

    jVijj2G2Ff2Mi jmMi jm24

    1 m

    2

    m2Mi j

    2;

    (2.12)

    where here Vij Vud for Mu d and Vus for Mus K. Since Mi j is a qi qj bound state, this rate is propor-tional to jc 0j2. With the normalization of c r in thenonrelativistic quark model determined by the conditionRd3rjc rj2 1, it follows that for a given 1S0 or 3S1 qi qj

    meson M,

    fM / jcM0jm1=2M

    : (2.13)

    Hence,

    jc K0jjc 0j

    fKf

    mKm

    1=2 2:3: (2.14)

    The difficulty of deriving this ratio from the NQM wasnoted early on as the van RoyenWeisskopf paradox[34].

    Conventionally, in the context of the quark model, the- mass difference was explained by means of a verystrong chromomagnetic, i.e., color hyperfine (chf) splittingbetween these particles. The similar, although smaller,mass difference between the K and K was also explained

    by this color hyperfine interaction. Taking account of color,the Hamiltonian for the color hyperfine (chromomagnetic)interaction has the form

    Hchf vchfrMiMj ~i ~j ~i ~j; (2.15)

    where the function vchfr involves the overlap of theinteracting constituent (anti)quarks. Recall that the product~i ~j is equal to 1 and 3 times the identity matrix I22when the qi and qj spins are coupled to S 1 and S 0,respectively. Similarly, the product of SU3c color matri-ces ~i ~j is equal to 16=3 and 8=3 times I33 if thecolors are coupled as 3 3 ! 1 (meson) and 3 3 ! 3(baryon), respectively. The resultant 1:3 ratio of massshifts in S 1 and S 0 q q mesons or quark pairs inbaryons and the 1:1=2 ratio of the color factor for q qmesons versus qq interactions in baryons yield good fits tomeson and baryon masses. The dependence of Hchf on1=MiMj is also important for this successful fit. In theNQM as applied here, vchfr / 3r, so that the colorhyperfine shifts evaluated to first order in Hchf are propor-tional to jc 0j2 (where the subscriptM on c is suppressedin the notation) [17]. This is analogous to the hyperfinesplitting in hydrogen, which is also proportional to thesquare jc 0j2 of the hydrogenic wave function at theorigin. We focus here on the 1S0 and

    3S1 isovector mesons,i.e., the and , absorb the color factor into the prefactorand thus write, for the energy due to Hchf ,

    Ehcf Ah ~i ~jijc 0j2

    MiMj: (2.16)

    The meson mass to zeroth order in Hchf is denoted m0.Then the physical masses are

    m m0 Ajc 0j2

    M2ud(2.17)

    and

    m m0 3Ajc 0j2

    M2ud: (2.18)

    Equivalently,

    m0 3m m

    4(2.19)

    and

    A m mM2ud

    4jc 0j2 : (2.20)

    Numerically,m0 620 MeV and the color hyperfine split-ting is

    Echf m m 4Ajc 0j2

    M2ud 640 MeV: (2.21)

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  • The color hyperfine interaction also plays an importantrole in splitting the K and , since their mass difference,mK m 360 MeV, exceeds, by about a factor of 3, thedifference in current-quark masses, ms md 120 MeV.In the NQM, this is attributed to the fact that the colorhyperfine interaction energy a=M2ud for the (with a >0) is negative and larger in magnitude than the correspond-ing energy a=MudMs for the K, since Ms >Mud.

    The large size of the splitting (2.21) shows that the colorhyperfine interaction is not a small perturbation on thezeroth-order Hamiltonian value, m0. Furthermore, thestrongly attractive, short-range color hyperfine interactionhas the effect of contracting the pion to a size substantiallysmaller than the size indicated by experimental data on thecharge radius and N scattering cross section. Moreover,since vchfr / 3r, which is clearly sensitive to short-distance interactions between quarks, there is a problem ofinternal consistency when one uses a color hyperfine in-teraction in the context of the nonrelativistic quark model,since at short distances, because of asymptotic freedom,the light quarks behave in a relativistic quasifree mannerwith their small, current-quark masses, not as nonrelativ-istic, massive, constituent quarks.

    The fact that Mud=Ms 0:7 has two countervailingeffects on the relative charge radii of the and K.First, since the reduced mass K is slightly greater than [cf. Eqs. (2.8) and (2.9)], it is plausible that the corre-sponding meson could be somewhat smaller. Yet the veryimportant attractive color hyperfine interaction should

    makeffiffiffiffiffiffiffiffiffiffiffiffiffihr2iK

    plarger than

    ffiffiffiffiffiffiffiffiffiffiffiffiffihr2ip

    .We proceed to discuss some properties of the color

    hyperfine interaction further. Since an attractive interactioninvolving a 3r function potential is inconsistent in thenonrelativistic constituent quark model, we model vchfras a spherical square well of depth V0 and radius r0. Thedimensionless quantity fixing the number of bound states isthe ratio of the strength V0 of a potential to the kineticenergy of a particle bound by this potential in a region ofsize r0, namely Ekin p2=2 2=2r20 (Ekin p =r0, relativistically). The ratio V0=Ekin 2V0r

    20=

    2. Since is determined by Eq. (2.7), we thusfix the product V0r

    20. One knows from general QCD theory

    that at distances r 1=QCD, the static quark potentialhas the Coulombic form

    Vq q C2fs

    rfor r 1

    QCD; (2.22)

    where C2f 4=3 is the quadratic Casimir invariant for thefundamental representation of SU3c and the logarithmicdependence of the running s on r is left implicit. Atdistances of order 1=QCD, Vq q has a linear form resulting

    from the chromoelectric flux tube joining the q and q,

    Vq q r for r 1QCD ; (2.23)

    where 1=20 400 MeV2 is the string tensionwith 0 the Regge slope. To illustrate the nature of the -puzzle, let us consider an infinite square-well potential,which provides a simple model of confinement. The (unit-normalized) ground-state wave function is

    c r

    2r30

    1=2

    sinpr

    pr

    ; (2.24)

    where p =r0. With this potential, one has

    hr2i Z

    r2jc rj2d3r 1

    3 1

    22

    r20; (2.25)

    so thatffiffiffiffiffiffiffiffihr2ip 0:532r0. The measured value of hr2i then

    determines r0 1:26 fm. Denoting hr2i d2, one canwrite jc 0j2 c=d3 with c a constant. In this model,

    c 2

    1

    3 1

    22

    3=2 0:236: (2.26)

    Substituting the value of jc 0j2 into Eq. (2.21), we find

    Echf 2AM2udr

    30

    (2.27)

    so that

    A m mM2udr

    30

    2 3:1: (2.28)

    Thus, both the large shift in Eq. (2.21) and the rather largevalue of the coefficient A show that the NQM treatment ofthe very strong, short-range hyperfine interaction as aperturbation is not really justified. As could be expectedon general grounds, such a strong short-range interactionhas the effect of producing a pion wave function that issmaller in spatial extent than is experimentally observed.In effect, the pionwhich, by definition, is the groundstate in the attractive 1S0 pseudoscalar channelis swal-lowed. i.e., squeezed into a contracted state of radiusmuch smaller than that of the . The wave function forthe itself slightly expands relative the original commonunperturbed and wave functions, due to the repulsivecolor hyperfine interaction in the 3S1 vector channel (whichis 1=3 as strong as the attraction in the 1S0 channel). TheNQM puzzle of a very light pion thus extends also to itsexpected much smaller size. In general, any extra, attrac-tive, potential that binds the pion more strongly than the yields a that is smaller than the . The only way tomaintain a common shape for the and wave functionsin a nonrelativistic potential model framework is to havevchfr constant as a function of r, which is very differentfrom the NQMs form vchf / 3r.

    III. A PHYSICAL PICTURE OFAPPROXIMATENAMBU-GOLDSTONE BOSONS

    NJL-type models do succeed in producing a massless ornearly massless pion in a bound-state picture, as was

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  • shown first via a solution of the Bethe-Salpeter equation inthe 1S0 channel in the original work by Nambu and Jona-Lasinio [14] (with an appropriate reinterpretation of thefour-fermion operator as involving quarks rather than nu-cleons in a modern context [23]). However, the coupling ofthe four-fermion operator is not calculated directly fromthe underlying QCD theory. Furthermore, this four-fermion operator posits that the spontaneous chiral sym-metry breaking is a contact interaction, and thus does notdirectly include the physically appealing mechanism forSSB as being a consequence of helicity reversal due toconfinement [1].

    An important insight for understanding the pion as a q qbound state and also an approximate Nambu-Goldstoneboson has been the argument by Brodsky and Lepagethat the physical pion state contains not just the valencejq qi state, but large contributions from higher Fock statessuch as jq q ngi, jq qq qi, jq qq q ngi (g gluon), etc.[35]. These higher Fock space states can account for muchof the size of the physical pion. This view is in accord withthe Goldstone phenomenon in condensed matter physics,where a Goldstone excitation is a collective state (e.g., aquantized spin wave or magnon in the case of a ferromag-net). This insight has been deepened with further workusing the light front formalism [36]. Recently, Brodskyand de Teramond have used AdS/QCD methods to calcu-late hadron masses including m, and also f, hr2i , andthe pion electromagnetic form factor Fq2 in the space-like and timelike regions [37].

    Another approach is provided by approximate solutionsto Dyson-Schwinger and Bethe-Salpeter equations. In ad-dition to phenomenological four-fermion NJL-type kernelsfor the Bethe-Salpeter equation [14,23], these have in-volved quark-gluon interactions in an effort to modelQCD [22,38]. These equations capture some of the relevantphysics, although they do not directly include effects ofconfinement or nonperturbative effects due to instantons.Confinement means that both quarks and gluons havemaximum wavelengths, i.e., minimum bound-state mo-menta, which affect chiral symmetry breaking [39].(Solutions of Dyson-Schwinger and Bethe-Salpeter equa-tions have also been used to investigate the dependence ofthe hadron mass spectrum on the number of light flavors ina general asymptotically free, vectorial non-Abelian gaugetheory [40].)

    Thus, there has been continual progress in understandingthe pion (and kaon) as both a q q bound state and anapproximate Nambu-Goldstone boson. Here we wouldlike to present a rather simple heuristic picture of thisphysics which, we believe, contributes further to thisprogress. For technical simplicity, we restrict ourselves tothe large-Nc limit, in which quark loops have a negligiblysmall effect. In this limit a simple proof that spontaneouschiral symmetry breaking occurs was constructed by show-ing that the t Hooft anomaly matching conditions [3] for

    massless u and d quarks must be realized in the physicalspectrum via a massless Nambu-Goldstone pion ratherthan massless nucleons [4]. To motivate our picture, wenote several elements that were missing in the nonrelativ-istic quark model approach to the - puzzle:(i) We need a natural mechanism for producing a suffi-

    ciently strong q q interaction in the 1S0 channel toreduce the mass of the bound state so that, up toelectroweak corrections, it vanishes in the limit ofzero current-quark masses.

    (ii) We would like the same physical picture to explainhow the pion is both a q q bound state and anapproximate Nambu-Goldstone boson whose mass-lessness follows from the spontaneous breaking ofthe SU2L SU2R global chiral symmetry downto the diagonal, vectorial SU2V and whose inter-actions involve derivative couplings, which vanishas q ! 0. As part of this, it is desirable that thepicture should yield the Gell-Mann-Oakes-Renner(GMOR) formula for the pion (and kaon) mass[41,42].

    (iii) We would like to resolve the van RoyenWeisskopfparadox [34] and explain how the quark wave

    function of the pion at the origin, c 0, can be /m1=2 and thus be consistent with a finite value of fin the chiral limit where m ! 0.

    (iv) Finally, we would like to understand how an ap-proximate Nambu-Goldstone boson such as thepion, which appears quite different from other had-rons, can have, as indicated by experiment, roughlythe same size as these other hadrons.

    We next present our new picture and show how itaddresses these questions. As is well known, if a quarkhas zero current-quark mass, the covariant derivative q 6Dqin the QCD Lagrangian, preserves chirality. A dynamical,constituent quark mass can be generated via an approxi-mate solution of the Dyson-Schwinger equation for thequark propagator. In the one-gluon exchange approxima-tion, one finds a nonzero solution for the effective quarkmass M if C2fs 4=3s * O1. In this framework,the dynamical quark mass M is thus the consequence of asufficiently strong quark-gluon coupling at the relevantscale, s at QCD. This dynamical quark masscan also be seen to result from the helicity reversal dueto confinement [1]. These two approaches can also be seento connect with the NJL-type analysis, with M Gh qqi h qqi=2f2, where G denotes the NJL four-fermion cou-pling. If one restricts oneself to a quenched approximationin which there are no quark loops, then the presence ofhigher Fock space states with q q pairs inside a meson withits valence constituent quarks can be regarded as being dueto a kind of zitterbewegungmotion of these valence quarks.Light-quark mesons which are not approximate NGBs,such as the , can be modeled satisfactorily as beingcomposed simply of a constituent quark and constituent

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  • antiquark. The effective size of this constituent q q boundstate is of order 1=Mud. The application of the nonrelativ-istic constituent quark model to such mesons is reasonable,with the constituent quarks moving in an approximatelynonrelativistic fashion under the assumed confining poten-tial, with a weakly repulsive hyperfine interaction for the meson.

    In the pion, however, the interaction at all scales is(strongly) attractive. This is manifested in the Euclideanpseudoscalar correlator. We recall that several generalproperties of hadrons have been understood on the basisof Euclidean correlation function inequalities [812]. Letus consider the Euclidean correlation function for a pseu-doscalar q q bound state,

    Px; y h ux5dx dy5uyi: (3.1)Performing the Gaussian fermionic (Grassmann) integra-tion in the path integral yields

    Px; y Z

    dAxSAyx; ySAx; y; (3.2)

    where dAx is the positive measure of the Euclideanpath integral, including the eSG factor from the gauge partof the action, where

    SG 14GG; (3.3)and the fermionic determinant is absent in the quenchedapproximation used here. In Eq. (3.2), SAx; y is thepropagator of the quark (a light u or d quark) movingfrom the initial position y to the final position x, in thepresence of the background gauge field Ax P

    aTaAax. SAyx; y denotes the Hermitian adjoint of

    SAx; y in color and Dirac space. We use the relation5SAy; x5 SAyx; y: (3.4)

    This property is unique to 5 and is not shared by any of theother 16 Dirac matrices. It ensures that the path integrand ispositive for all field configurations, making Px; y largerthan all other Euclidean (scalar, vector, axial-vector, andtensor) correlators. Asymptotically, when jx yj ! 1,any correlatorCx; y behaves, up to a power-law prefactor,as

    Cx; y expm0jx yj; (3.5)where m0 is the mass of the lightest physical state with thequantum numbers of the correlator considered. This, to-gether with the inequality

    Px; y expmjx yj any Cx; y; (3.6)guarantees that the pion is, indeed, the lightest meson.Furthermore, the positivity of Px; y for any jx yj andthe fact that Sx; y is a monotonically decreasing functionof jx yj implies that the effective quark-antiquark poten-tial in the pion (to the extent that this nonrelativistic

    language is appropriate) is attractive at all relativedistances.As we noted in the previous section, in the nonrelativ-

    istic quark model a very strong hyperfine interaction be-tween the quark and antiquark in the pion is needed inorder to reduce its mass nearly to zero, and such aninteraction tends to produce a wave function for the va-lence q q in the pion that is restricted to a very small spatialextent (almost collapsed). Following this lead, we suggestthat , while the spacetime (or Euclidean) picture of a qqvector meson is two wool-ball-like single strands of va-lence quark and antiquark lines, the pion is a double strand,namely, closer valence q and q world lines whose motionforms a single wool-ball-like configuration. According tothis picture, in the pion, but not in the etc., the valence qand q lines with collinear momenta track each other at adistance that is shorter than 1 fm. This is similar to NJL-type models, in which, by construction, the importantinteraction is of short range. Thus, in our picture the pionqualitatively differs from the as a nearly Nambu-Goldstone particle should and, at the same time, can beconsistently considered as a qiqj state. Here and below,

    analogous comments, with obvious changes for the heavierms, apply for the K and its comparison with the K

    .It is well known from discussions of the chiral anomaly

    [49] that a massless collinear quark and antiquark ofopposite helicity, corresponding to the bilinear operatorproduct c5c , can mimic the pole of a massless pseudo-scalar particle and replace the latter in the calculation ofthe anomaly. Here we suggest that such a configuration,including the effect of spontaneous chiral symmetry break-ing, can represent the massless pion, explain the puzzlingstrong color hyperfine interaction between the qi and qj,

    and can account for its behavior as a light, approximateNambu-Goldstone boson. In general, one would expectfrom the basic quantum mechanical relation piri *@ that restricting the q and q to a small interval along someaxis xi would entail large momenta along this axis.However, in the presence of an appropriate gluonic fieldconfiguration, the gauge-covariant momentum pIgTaAa can vanish.

    We address the questions posed above, starting with (i).There are two sources of explicit chiral symmetry break-ing, namely, finite quark masses and the presence of non-zero electroweak interactions. For our present discussionwe shall imagine that, unless otherwise indicated, electro-weak interactions are turned off. Then a nonzero pion massis induced via the explicit chiral symmetry breaking termmu ddmu uu in the QCD Lagrangian. To see this in ourpicture, we consider the spacetime evolution of the qi andqj in a pion, after a Euclidean Wick rotation. In the QCD

    context, the qi and qj are connected by a chromoelectric

    flux tube, and their positions fluctuate within length scalesof order 1=QCD 1 fm. We consider a model in which inthe pion, but not in the or other vector mesons, the

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  • valence q and q track each other at a distance h shorter than1=QCD, at all times. The constituent quark masses arethen less relevant to the dynamics, being gradually re-placed, as h gets smaller, by their current-quark masses.The point here is that constituent quark masses are con-sequences of spontaneous chiral symmetry breaking,which disappears at short distances (large momenta). InQCD, the scale-dependent dynamically generated constitu-ent quark mass decays, as a function of Euclidean mo-menta p, like

    Mq h qqip2

    (3.7)

    up to logs [22], where

    h qqi 4f3 3QCD: (3.8)More generally,

    Mq QCDQCDp

    2

    ; (3.9)

    where denotes the anomalous dimension of the qqoperator; here we use the property that is a power seriesin the running coupling s, ands approaches zero at shortdistances because of the asymptotic freedom of QCD. Inour picture it is this melting away of the constituentquark masses at short distance which provides, in the NQMlanguage, the very strong hyperfine interactions in the pion.

    Next, as an answer to question (ii), we would like toshow how the GMOR relation for the pion and otherpseudoscalar meson masses (aside from the 0), whichembodies the Nambu-Goldstone nature of these pseudo-scalar mesons, is naturally expected in our picture. Let the

    total Euclidean length R jxj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 r2

    p, where it,

    be the net Euclidean distance traveled by the q q doubleline describing the valence quark-antiquark in the pion.The double line describes a random walk with n straightsections of total length L. When probed at distances thatare short compared with 1=QCD, the quark masses are thehard, current-quark masses, mu 4 MeV and md 8 MeV. When mu mdjxj * 1, the quark propagationinvolves the suppression factor emumdjxj. Hence, a char-acteristic length describing this propagation is

    L / 1mu md : (3.10)

    Now the end-to-end distance R for a randomwalk with stepsizes d (in any dimension) is given by [43]

    R2 d2n: (3.11)On average, if the total length of the n-step walk is L andthe step length is d, then

    d Ln: (3.12)

    Hence,

    R2 Ld: (3.13)Then the basic correlation function relation, Eq. (3.5),implies that

    m 1R (3.14)

    and hence that

    m2 mu mdd : (3.15)

    Since the step size d is connected, via the helicity reversalprocess, to the underlying confinement and dynamicalbreaking of chiral symmetry, it is natural to equate

    1

    d h qqi

    f2(3.16)

    (where we follow the usual phase convention for the quarkfields so that, with mq taken as positive, the condensate

    h qqi< 0). Combining these with Eq. (3.15), we see thatthis heuristic analysis yields the GMOR mass relation,

    m2 mu mdf2

    h qqi; (3.17)

    where hqqi hPNca1 qaqai with q u or q d (thesecondensates being essentially equal in QCD). A similarargument, with appropriate replacement of light-quarkmass mu or md by ms, yields the analogous GMOR-typemass relations for the K and K0,

    m2K

    mu msf2K

    h qqi (3.18)

    and

    m2K0

    md msf2K

    h qqi; (3.19)

    where h qqi h ssi for q u, d.The nearby q and q paths in our picture generate q q

    color interactions that depend on the difference of thesepaths. This suggests the possibility of using this picture toinfer the derivatively coupled form of pion interactionsappropriate for a Nambu-Goldstone particle. This deriva-tive coupling means that in the static limit, these Nambu-Goldstone bosons become noninteracting.We next address point (iii) above, concerning the rela-

    tion f jc 0j= ffiffiffiffiffiffiffimp . Since the matrix element (2.10) asit enters in the ! decay amplitude obviouslyinvolves the annihilation of the u and d quarks in the to produce the virtual timelike W that, in turn, producesthe pair, it clearly depends on the u dwave function inthe pion evaluated at the origin of the relative coordinate,jc 0j. The question here concerns what happens in thechiral limit, where m ! 0. For this discussion we again

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  • imagine that electroweak interactions are turned off, exceptthat we take into account the couplings leading to the 2decay. Now the pion wave function at a given time involvesthe intersection of the worldlines of its constituent q and qwith the t 0 hyperplane in the full R4 Wick-rotatedspacetime. This wave function has many Fock space com-ponents. The matrix element (2.10) involves the annihila-tion of the valence qi qj component by the axial-vector

    current. Higher Fock space components in the pion wavefunction correspond to additional crossings of the t 0hyperplane. A measure of the contributions of these addi-tional components can be obtained from our random walkrepresentation. We note that for the present purpose it isessentially a one-dimensional random walk that is relevant,since we are inquiring about passages across a hyperplane,namely, that defined by the condition t 0, of codimen-sion 1 in the full EuclideanR4. Now in general, the numberof times that a one-dimensional random walk with n stepsreturns to the origin is asymptotically / ffiffiffinp for large n.The contribution of the valence q q component of the fullpion wave function to the annihilation probability jc 0j2is thus reduced by the factor 1=

    ffiffiffin

    p. By Eq. (3.11), n1=2 /

    R1 and by Eq. (3.14), R1 m, so jc 0j2 is reducedby the factor m. This means that jc 0j / ffiffiffiffiffiffiffimp in thechiral limit, thereby canceling the

    ffiffiffiffiffiffiffim

    pin the denominator

    of Eq. (2.13), and yielding a finite value of f. Similarremarks apply for fK in the hypothetical limit ofms ! 0 aswell as mu;d ! 0. Thus, our picture provides a plausibleresolution of the van Royen and Weisskopf paradox[Eq. (2.13)] [34].

    Finally, we address issue (iv) concerning the similar sizeof the and . We should emphasize from the very outsetthat this is challenging. The qualitatively different physicalpictures involved give an indication of the complexity inthe calculation of charge radii. On the one hand, if the sizeis controlled by the relatively small separation h in ourpicture with double qi qj lines, then the pion should be

    much smaller than the . On the other hand, since thedistance R 1=m controls the overall pion size, it followsthis size can become, at least formally, unbounded in thechiral limit m ! 0. (In practice, pion wave functionscentered within a distance R of each other would overlapand become entangled.) This divergence in R asm ! 0 isnot an artifact of our picture; the range of the residualstrong force mediated, at long distance, by pion exchange,formally diverges in this chiral limit. The property that thepion charge radius also diverges in the chiral limit is anatural concomitant of this divergence in the pion size. Letus elaborate on this.

    The charge radius (squared) of a hadron is

    hr2i Z

    rr2d3r; (3.20)

    where r denotes the charge density. The quantity ffiffiffiffiffiffiffiffiffiffiffijhr2ijpgives one measure of the size of a composite particle [44].

    This is especially clear for a meson such as the or K,where the u and, respectively, d or s both contributepositively to the integrand in Eq. (3.20) [44,45].The charge radius squared is proportional to the slope of

    the electromagnetic form factor Fq2 at q2 0 [46]:

    hr2i 6dFq2

    dq2

    q20: (3.21)The latter form factor satisfies a t-channel dispersion

    relation (t q2)

    Ft Z

    dtImFt0t t0 : (3.22)

    In particular, for the case under consideration, Ft Ft, the integration is from t0 2m2 to t0 1. Inthe vector meson dominance approximation for Ft, onecommonly replaces the ImFt0 by a delta function cor-responding to the approximation of zero width for therelevant vector meson. Here, using dominance forFt, one replaces ImFt0 by a delta function/ t0 m2. This narrow-width approximation, togetherwith the known value F0 1, yields

    Fq2 m2

    m2 q2; (3.23)

    so that

    ffiffiffiffiffiffiffiffiffiffiffiffiffihr2i

    q

    ffiffiffi6

    pm

    0:62 fm: (3.24)

    This is close to the experimentally measured value, givenabove in Eq. (2.1) [33,37]; quantitatively, it is smaller thanthis experimental value by only 7%. (One can also includethe effects of the width, but this will not be necessary forour discussion here.) An analogous vector meson domi-nance prediction for the K charge radius works very wellalso [33]. A priori, one might worry that an additionalthreshold contribution from t0 2m2 might dominateand lead to hr2i 1=m. However, this does not happenhere because of the derivative coupling of soft pions, asNambu-Goldstone bosons. In the particular case here, an-other reason why this does not happen is that there is affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit0 4m2

    pfactor in ImFt0 that arises from the P-wave

    nature of the amplitude. Nevertheless, hr2i doesdiverge as hr2i ln1=m in the chiral limit wherem ! 0 [47].We next sketch an estimate of the pion charge radius in

    our picture. As in the previous section, the higher Fockspace states play a key role in this estimate. Consider thet 0 slice of the Wick-rotated Minkowski space. Thequantity hr2i can be computed as a sum of the contribu-tions of the various Fock space components of the pionwave function. In our picture these are generated by cross-ings of the t 0 hyperplane by the qi qj random-walkingdouble worldline of the pion. Let the kth such crossing be

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  • at rk. The first crossing corresponds to a , say, moving

    forward in time. Hence, we have a charge 1 at thislocation. At the second crossing, the pion line is reversed,and we have a 1 charge at r2, etc. The definitionEq. (3.20) above then yields hr2i

    P1k11kr2k, where

    here rk jrkj. Recalling that the kth visit to the t 0plane happens typically after n k2 steps of the random-walking double line, with individual step size d, we deducethat, on average, r2k k2d2. By itself, this would yield, forhr2i , the sum

    P1k11kk2d2. The terms in the above

    oscillating sum diverge as k ! 1. However, to get theactual sum, we must take into account the fact that thecontributions are regularized by the exponentialexpmu mdL expmu mdk2d controllingthe total length of the random-walking double line. Usingmu md dm2 from Eq. (3.15) above and defining

    b md; (3.25)we can rewrite the charge radius as

    hr2i X1k1

    1kk2d2ek2b2 : (3.26)

    Since k gets very large in the chiral limit, there are strongcancellations between successive terms, rendering an ac-curate estimate difficult. We can at least investigate thenature of the leading divergence in hr2i . To do this, wereplace the above sum, after subtracting and adding an r0term and symmetrizing, by an integral over the variable

    kb kmd:

    Ib 1m2

    Z 11

    d

    2 exp

    i

    b 2

    ffiffiffiffi

    p2m2

    1

    ffiffiffiffi

    p2md2

    exp

    2md

    2: (3.27)

    The key observation here is that, while we have, as ex-pected, an explicit 1=m2 factor in front, the integral Iband any finite derivative thereof, contain the factorexp2=2md2, which vanishes with an essentialzero in the chiral limit m ! 0. This can be seen as aconsequence of the strong cancellations between differentterms contributing to the sum, which we approximated asan integral. Thus, our calculation shows the absence of adivergence of the power-law form 1=m2

    in hr2i and isconsistent with the chiral perturbation theory result thathr2i diverges like ln1=m in this limit. For the realworld with nonzero current-quark masses for u and d, ouranalysis above naturally yields a value of hr2i d2,since this was the r0 term in the sum. A similar conclusion,with appropriate replacement of the d with the s quark,applies to hr2iK in the SU(3) chiral limitmu, md, ms ! 0.

    Our picture can also give a plausible explanation of whythe pion-nucleon cross section N at energies above theresonance region can be comparable to the inferred valueof N at the same energies [cf. Eqs. (2.4) and (2.6)].

    Relevant to theN cross section is the fact that the valencequarks in the pion propagate in an extended double-linemanner covering an area of order R2 1=m2. However,because of the strong color hyperfine interaction, the sepa-ration h of the valence qi and qj in the pion is rather small

    in our model. Hence, while in a crossing of two qi qj pairs

    at an ordinary hadronic distance d, the probability of aninteraction is O1, here, in contrast, it will be Oh=d2.In the context of a hadronic string picture, the small pionmass is related to the separation h viam / h, where isthe hadronic string tension. Hence, one may roughly esti-mate that the N cross section N contains the factorsR2h=d2 / =d2. Note that the factor of m2 can-cels out between numerator and denominator, leaving Nproportional to an expression involving the string tensionand a typical hadronic distance scale, which are the samefor the and the . In the preceding we have presented ourefforts to show how our picture of a rather tightly boundqi qj pair undergoing a random walk inside a pion can

    explain how this particle can exhibit the properties of anapproximate Nambu-Goldstone boson while also beingunderstandable as a qq bound state. Ultimately, one shouldbe able to find the differences predicted by our picture ascompared with other approaches to this physics. One theo-retical tool that is relevant here is lattice gauge theory.However, one faces not only the technical difficulty ofsimulating very light quark masses and light pions. Anadditional challenge is that in (Euclidean) lattice simula-tions one first integrates over the fermionic degrees offreedom. Having the two (say u and d) quark propagatorsin the same background color field may not allow one toverify that at all intermediate steps the quark and antiquarkare really close to each other. One may need to go back tothe sum over fermionic paths in order to actually detect thepropagators of the nearby qi qj pair.

    One implication of our model with the nearby qi qj lines

    separated by a relatively small distance h is that the purelygluonic exchange amplitude for scattering should berather small. A recent lattice calculation of the I 2S-wave scattering length obtained the result a2 0:043=m [48], in agreement with the Weinberg-Tomozawa soft-pion current algebra result a2 m=16f2 0:044=m [49,50]. In a hypothetical0 scattering, where the 0 is comprised of d0 and u0quarks that are degenerate with the ordinary u and d but donot mix with them, the scattering amplitude involves onlygluon exchanges, but not quark interchanges. In this case apreliminary lattice calculation has obtained a 0 scatter-ing length considerably smaller than a2, in qualitativeagreement with our discussion above [51].

    IV. SOME COMMENTS ON THE K ! ANDHEAVY-QUARK TRANSITION FORM FACTORS

    TheKmesons undergo semileptonicK3 decays, such asK ! 0, K0L ! , and K0L ! , me-

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  • diated by the vector part of the weak charged current. Thealmost conserved vector current (CVC) [conserved apartfrom SU(3) flavor-breaking effects] helps to fix the corre-sponding hadronic matrix elements and the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix elementjVusj [52] and tests of three-generation CKM unitarity[53]. Denoting the 4-momenta p pK p and q pK p, one has

    h0jVjKi hjVjK0Li

    1ffiffiffi2

    p fq2p fq2q; (4.1)

    where V is the weak charged current. The contributionfrom the fq2 term is proportional to the final leptonmass squared and is negligible for semileptonic decays tofinal electrons. The t q2 variation of ft over the rangem2e t mK m2 can be approximated by a linearfunction of t:

    ft f01 t

    m2

    ; (4.2)

    where 0:0288 [20], in agreement with chiral pertur-bation theory calculations [47,54] and also with the expec-tation from simple K vector meson dominance. CVCimplies that fq 0 1 in the hypothetical limit ofexact SU3V symmetry, mu md ms and hence mK m. A general result which wewill elaborate on later is thatthe corrections to this symmetry-limit value are alwayssecond order in SU3V breaking. This is the well-knownAdemollo-Gatto theorem [31]. This feature is evident inthe explicit estimate [55,56]

    f0 1 5m2K m22

    3842f22m2K m2 0:985: (4.3)

    The correction term in Eq. (4.3) arises from multiparticlecontributions to the sum-rule corresponding to the commu-

    tator Q4i5; Q4i5 Q3 ffiffiffi3

    pQ8, where the subscripts

    refer to SU(3) flavor generators. Formally, this sum ruleunderlies the Ademollo-Gatto theorem; the multiparticlecontributions are squares of matrix elements of the diver-gence @J

    4i5 of the strangeness-changing weak vector

    current, making the deviation from universality quadraticin the SU3V symmetry breaking. However, as is alsoevident in (4.3), in the limit of SU3L SU3R chiralsymmetry, with m ! 0 and mK ! 0, the correction termis actually of first order [55]. This is a consequence of thefact that in this chiral limit the contributions to the sum rulethat involve the exchange and propagation of massless sand Ks lead to a deviation from universality that is linearinm2K. The correction in Eq. (4.3) is small partly because ofthe numerical coefficient arising from the loop diagraminvolved in the calculation.

    It is instructive to see how the Ademollo-Gatto theoremis realized in the NQM, where the form factor fq2 can

    be expressed as an overlap of the K and wavefunctions, which we shall denote FK!q2. In the NQM,the andK consist of nonrelativistic constituent quarksqi qj and for L 0,

    FK!q 0 Z

    d3rc rc Kr: (4.4)

    In this model the K and wave functions (which are real)depend on just an overall flavor-independent potential Vrand on the reduced masses K and . We recall ournotation Mq for the constituent mass of a quark q, and

    the valuesMu Md Mud 330 MeV,Ms 470 MeV.For simplicity we use the single-term form for the

    potential:

    Vq qr V0r

    r0

    ; (4.5)

    where is an exponent. Special cases include (i) 1,i.e., Coulombic, (ii) 0, with Vr / lnr; (iii) 1,linear; (iv) 2, harmonic oscillator; and (v) 1,equivalent to an infinite square-well potential. As wasnoted above, a realistic quark-quark potential has differentforms at short distances and at distances of order1=QCD 1 fm, so it is more complicated than a single-term form. However, the simplification will suffice for ourpurposes here. The scaling properties of the Schrodingerequation imply that the spatial extent r characterizing thefalloff of the wave function scales with the reduced massas [18,57]

    r / 1=2: (4.6)For the range of considered here, the dependence of thischaracteristic distance on is thus maximal for theCoulombic, 1, case and minimal for 1, wherethe spatial extent of c is determined completely by thewidth of the infinite square well and is independent of .For 2, i.e., the harmonic oscillator potential, which

    we write as V kr2=2, the wave function is proportionalto a Hermite polynomial, and, for the ground state, it is

    c k3=8

    3=2exp

    ffiffiffiffiffiffiffik

    pr2

    2

    : (4.7)

    Substituting this into Eq. (4.4), we calculate

    FK!0 23=2K3=8

    ffiffiffiffiffiffiffiKp ffiffiffiffiffiffiffip 3=2 : (4.8)

    Let us define the following measure of flavor SU(3) sym-metry breaking:

    Ms MudMud

    : (4.9)

    The expression for FK!0 in Eq. (4.8) has the followingTaylor series expansion in :

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  • FK!0 1 3256 2 O3 for 2: (4.10)

    With the values of and K given above,

    FK!0 0:999 for 2: (4.11)For comparison, consider the Coulomb potential withground state

    c er=a

    1=2a3=2; (4.12)

    where

    aB 1C2fs (4.13)

    is the Bohr radius and C2f 4=3. Substituting this intoEq. (4.4) for the wave function overlap, we find

    FK!0 2

    ffiffiffiffiffiffiffiffiffiffiffiffiaKa

    paK a

    3

    2

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK

    pK

    3 0:992:

    (4.14)

    This again has a Taylor series expansion of the formFK!0 1O2, as expected from the Ademollo-Gatto theorem. Since the r dependences of the logarithmic( 0) and linear ( 1) potentials are intermediatebetween the harmonic oscillator ( 2) and Coulomb( 1) potentials, one expects Fq 0 to be veryclose to unity for these potentials as well.

    These results do not imply such small deviations fromunity for the form factor f0 in Ke3 decay. The massdifference mK m 360 MeV far exceeds the value ofms mu expected in a model with a flavor-independentconfining potential. Such considerations would apply bet-ter to semileptonic s ! u decays of mesons with heavy cor b spectator quarks. Indeed mDs mDu 104 MeV andmBs mBu 89 MeV, consistent with the current-quarkmass difference ms mu. Unfortunately, these small massdifferences imply tiny branching for these decays Ds !Du

    and Bs ! Bu.The generic form factor is a Lorentz-invariant function

    of q2. However, for elastic scattering, one can go to a framewhere q0 0 so that q2 jqj2 and write

    Fq2 Z

    d3reiqrc rc Kr; (4.15)

    where q q0;q is the momentum imparted to the lep-tons in the decay process, r rq r q, and c K and c arethe initial and final meson wave functions. In the flavorSU(3) symmetry limit, c K c . For q 0, the normal-izations of the wave functions imply the conserved vectorcurrent (CVC) value F0 1.The last result is quite general; if the mesons contain, in

    addition to the valence quarks qi and qj, any number of

    gluons at the position Rs and/or q q quark pairs at thepositions r, r0 , we would have, instead of (4.15),

    FK!q2 Z

    d3reiqrY;0;s

    d3rd3r 0d

    3Rs

    c r; r; r 0 ;Rsc Kr; r; r 0 ;Rs;

    (4.16)

    so that again in the flavor SU(3) symmetry limit, for equalwave functions and q 0, we have F0 1. Here, c Kand c are the Fock space wave functions with anynumber of gluons and quark-antiquark pairs. Both quarksand gluons carry spin and color, so that c could be asuperposition of many color and spin couplings whichyield overall color singlets. For notational simplicity wehave omitted these above. The general arguments pre-sented below do not depend on the slightly simpler form of(4.16).As is evident in Eqs. (4.15) and (4.16), deviations from

    F0 1 can be caused in two ways. First, even for elastictransitions with c initial c final, the momentum transferfactor eiqr modulates the positive integrand and decreasesF. Second, flavor SU(3) breaking, namely, the differencebetweenms andmq, q u, d, causes the and K wavefunctions to be different and hence reduces f0 fromunity. To analyze this, we shall use the Cauchy-Schwarzinequality, that for any vector spaceV with vectors c and and an inner product hc ; i, the property

    jhc ; ij kc kkk (4.17)

    holds, where kc k ffiffiffiffiffiffiffiffiffiffiffiffiffiffihc ; c ip . We apply this to the L2Hilbert space of square-integrable functionsc r; r; r 0 ;Rs with the inner product

    hc ; i Z

    d3r

    Y;0;s

    d3rd3r 0d

    3Rsc r; r; r 0 ;Rsr; r; r 0 ;Rs: (4.18)

    Thus,

    FK!q 0 hc ; c Ki: (4.19)

    Using this, we have

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  • jFK!q 0j2 Z

    d3r

    Y;0;s

    d3rd3r0d

    3Rsc r; r; r 0 ;Rsc Kr; r; r 0 ;Rs

    2

    Z

    d3r

    Y;0;s

    d3rd3r0d

    3Rsjc r; r; r 0 ;Rsj2Z

    d3rY;0;s

    d3rd3r0d

    3Rsjc Kr; r; r 0 ;Rsj2

    ; (4.20)

    where we write these for the general case of Eq. (4.16)above.

    Universal form factors at the no-recoil point for semi-leptonic decays of mesons containing heavy quarks, e.g.,Bd ! D, follow from the fact that for mb >mc QCD, the heavy quark is a static source of color [trans-forming as a color SU(3) triplet] with common wavefunctions for all of the light degrees of freedom in eitherthe B or D mesons [58]. With flavor-independent primaryQCD interactions, the difference between c K and c isdue to the different u and smasses only. From the Cauchy-Schwarz inequality, one sees that Fq 0, as a functionof ms and ms mu, is extremal (maximal) at ms mu 0. Hence, the deviation from unity is of order Oms mu2, which is the Ademollo-Gatto theorem [31]. Theanalogous theorem for heavy quarks is derived by thesame type of reasoning [59,60]. Let us rewrite (4.20) as

    FB!Dq 0milight; v mi=MQ; (4.21)where milight refers to the masses of the degrees offreedom that are light relative to mQ, namely QCD, ms,etc. and MQ denotes the mass of the lighter among theheavy quarks, namely mc in the present case. Again, F isextremal for v 0, and the deviations from universality atthe no-recoil point are of order Ov2 O1=m2Q, i.e.,O1=m2c in b ! c transitions. Indeed, if the current-quarkmasses mu md 0, then, when ms ! 0, the chiral sym-metry group is enlarged from SU2L SU2R toSU3L SU3R [and the QCD condensates would thenbreak these to the respective diagonal subgroups SU2Vand SU3V].

    The generalized Ademollo-Gatto theorem can be for-mulated in Hamiltonian lattice QCD. The and K wavefunctions are replaced by wave functionals with arbitrarypatterns of excited links, corresponding to gluonic excita-tions, and/or extra q q pairs. Now consider the matrixelement of the strangeness-changing vectorial weakcharge, Qu;s

    Rd3xV0, where V denotes the associated

    current. This converts flavors s ! u for the valence quarks.Since Q;H 0, this changes the energy of the stateoperated on by mK m. However, since Q is an integralover all space, it does not change the 3-momentum of thestate on which it operates. Hence, if it operates on a K atrest, it should produce a at rest also. The matrix elementof interest is the overlap of two wave functionals computedfor valence quark mass mq ms and for mq mu. By theCauchy-Schwarz inequality, which holds for these wavefunctionals, the overlap is smaller than unity, achieving its

    maximum value when ms mu 0. Hence, repeat-ing the same arguments as above, we find that

    Fq 0 1O2: (4.22)

    V. MASS COMPARISONS INVOLVING HEAVIERHADRONS

    We proceed to discuss the systematics of mass differ-ences mQs mQ u for various JPC mesons. Somerelated work is in Refs. [32,61]. In this context, we recallthat modern lattice estimates have yielded a somewhatsmaller value of the current-quark mass ms 120 MeVthan some older current algebra estimates, which tended tobe centered around 180 MeV [47]. In the nonrelativisticquark model [with a flavor-independent nonrelativisticquark-(anti)quark interaction potential], the mass differ-ence between analogous hadrons differing only by havingan s quark replaced by a u or d quark should, up to smallbinding changes due to the different reduced constituentmasses, differ by ms mud. The real world is more com-plicated, for several reasons. First, the concept of quarkmasses and differences needs to be carefully defined. Themasses run with the distance or momentum scale at whichthey are probed. The constituent quarks can be consideredto be extended quasiparticles, confined to hadrons withsizes of order 1=QCD. As the MIT-SLAC deep inelasticscattering experiments showed dramatically, as one in-creases the momentum scale at which one probes such aquark beyondQCD, it acts quasifree, without the attendantstrong coupling to gluons to which it is subject for mo-menta less than QCD. As this momentum scale increasesconsiderably beyond QCD, the quark mass then goes overto approximately the current-quark mass, since the QCDgauge coupling becomes small. Since different hadronshave somewhat different effective scales, this modifiesthe extracted mass difference.Second, while at the fundamental Lagrangian level the

    only breaking of flavor symmetry is due to the differencesbetween the current quark masses, this is not the case forthe effective potential between the constituent quarks in thenaive quark model because of the short-range color hyper-fine interactions, though not in the asymptotic, confiningpart of the potential. This suggests that the mass differ-ences of Qs and Q q mesons with Q a heavy-quark betterestimate the current-quark mass difference ms mq massdifference, with q u or d, since both the magnitude ofthe color hyperfine splittings and the effective sizes of thesystem are smaller there (the latter is a reduced mass

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  • effect). Some measured mass differences, averagedover isospin multiplets, are mK m 120 MeV,m mK 125 MeV, mDs mDu mDsmDu 100 MeV, and mBs mBu 90 MeV. Weobserve a substantial and fairly systematic tendency ofthese mass differences to decrease as mQ increases.This is in agreement with the lattice gauge theory estimatesmentioned above. The pattern in the baryonic spectrum ismore complicated, but does not disagree with this generaldecreasing behavior. As is well known, the large splittingsin the JP 1=2 baryon octet, viz., m mN 180 MeV, m mN 255 MeV, m m 200 MeV, and m m 125 MeV, can be ex-plained by a color hyperfine interaction, similar to thatfor the mesons. The equal-spacing mass difference rulein the J 3=2 baryon decuplet with the interval of146 MeV can also be explained by the color hyperfineinteraction. The relatively large mass difference betweencsu; 1=2 c and cud; 1=2 c of mc mc 181 MeV is again in agreement with the expec-tation based on the large difference in the s u and d ucolor hyperfine interaction, which is evidently not reducedby the presence of the nearby heavy c quark in thesebaryons. Only the difference of masses of c css; 1=2 and c csu; 1=2 of 230 MeV appearsto be somewhat high. On the basis of this discussion, oneexpects small mass splittings mQQ0s mQQ0u be-

    tween baryons containing two heavy quarks, but this ex-pectation cannot yet be checked.

    VI. CONCLUSIONS

    In conclusion, we have revisited the - puzzle, namely,the problem of describing the meson as a qi qj bound

    state and as an approximate Nambu-Goldstone boson andrelating its mass and size to those of the meson. We havepresented a simple heuristic picture that, we believe, givesinsight into this problem. In this picture, the valence qi andqj quarks in the are rather tightly bound by the strong

    color hyperfine interaction that splits the and masses.We show that this picture can resolve another old puzzleconcerning the pion wave function at the origin (vanRoyenWeisskopf paradox) and is consistent with theGell-Mann-Oakes-Renner relation, With appropriate re-placement of the u or d quark by the s quark, our picturealso applies to the K and its relation to the K. Using ourmodel, we present an estimate for the charge radius hr2i .Our approach gives further insight into the charged-currentK transition relevant in K3 decays.

    ACKNOWLEDGMENTS

    S. N. thanks A. Casher and M. Karliner, and R. S. thanksS. Brodsky, for helpful discussions. The research of R. S.was partially supported by Grant No. NSF-PHY-06-53342.

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  • [44] Some illustrative semiclassical examples with total chargeQ Rrd3r 1 and positive-definite r which haveconstant, exponential, and Gaussian forms, respectively,are the following: (i) if r 3=4R3 for r R andr 0 for r > R, then hr2i 3=5R2; (ii) if r 8R31er=R, then hr2i 12R2; and (iii) if r ffiffiffiffip R3er=R2 , then hr2i 3=2R2.

    [45] The connection betweenffiffiffiffiffiffiffiffiffiffiffijhr2ijp and the size of the hadron

    is less direct for a particle in which the quarks contributewith opposite signs to the integrand of Eq. (3.20). Indeed,because hr2i flips sign under charge conjugation, hr2i 0for a self-conjugate particle, such as 0. However, isospininvariance implies that the size of 0 is equal, up to smallcorrections, to the size of . For an electrically neutral,non-self-conjugate meson Mi j, the sign of hr2i is con-trolled by the lighter of the qi and qj. For example,hr2iK0 0:077 0:010 fm2 [20], as can be under-stood from the larger spatial extent of the d comparedwith the s, which results from the property thatMud

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