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)