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
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 , 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 , 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 . 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 . 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  and Banks andCasher , 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., ) 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 . 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  (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
PHYSICAL REVIEW D 79, 016005 (2009)
1550-7998=2009=79(1)=016005(15) 016005-1 2009 The American Physical Society
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 , 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 . 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 
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 
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
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
SHMUEL NUSSINOVAND ROBERT SHROCK PHYSICAL REVIEW D 79, 016005 (2009)
of the constituent quark masses in these formulas dependsomewhat on the method that one uses to infer their values. 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 ,
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
jVijj2G2Ff2Mi jmMi jm24
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
fM / jcM0jm1=2M
jc K0jjc 0j
1=2 2:3: (2.14)
The difficulty of deriving this ratio from the NQM wasnoted early on as the van RoyenWeisskopf paradox.
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) . 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
The meson mass to zeroth order in Hchf is denoted m0.Then the physical masses are
m m0 Ajc 0j2
m m0 3Ajc 0j2
m0 3m m
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)
AND K AS q q BOUND STATES AND . . . PHYSICAL REVIEW D 79, 016005 (2009)
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