Are There Oscillations In The Baryon/Meson Ratio?

arXiv:hep-ph/9207242v1 15 Jul 1992McGill/92-07hep-ph/9207242March 1992Are There Oscillations In The Baryon/Meson Ratio?Jean-Ren´e Cudell(a) and Keith R. Dienes(b)Dept. of Physics, McGill University 3600 University St., Montr´eal, Qu´ebec, CanadaH3A-2T8All available data indicate a surplus of baryon states over meson states for energiesgreater than about 1.5 GeV.

Since hadron-scale string theory suggests that theirnumbers should become equal with increasing energy, it has recently been proposedthat there must exist exotic mesons with masses just above 1.7 GeV in order to fillthe deficit. We demonstrate that a string-like picture is actually consistent with thepresent numbers of baryon and meson states, and in fact predicts regular oscillationsin their ratio.

This suggests a different role for new hadronic states.Typeset Using REVTEX1

In a recent work, Freund and Rosner [1] have examined the separate densities of observedmeson and baryon states as functions of their masses. They find that the integrated num-ber of baryon states is less than that of meson states for masses less than about 1.7 GeV,but then greatly surpasses the meson number at higher energies.

Since hadron-scale stringtheories are successful in modelling not only the hadronic Regge trajectories but also theexponential (Hagedorn) growth [2] in the total hadronic density, Freund and Rosner pointout that such theories may also serve as the basis for understanding the relation betweenthe separate meson and baryon densities. This is possible in part due to a recent result ofKutasov and Seiberg [3] which states that the numbers of bosonic and fermionic states ina non-supersymmetric tachyon-free string theory must approach each other as increasinglymassive states are included.On the basis of this theoretical result, Freund and Rosnerpredict that there must exist a number of mesons yet to be discovered with masses above1.7 GeV (in order to match the rise in baryon number); furthermore, since the presently-observed baryon/meson ratio is consistent with quark-model calculations which include onlyconventional mesons and baryons [4] (i.e., states with qq and qqq quark configurations re-spectively), they additionally speculate that these new mesons are likely to be exotic (withquark content qp+1 qp+1, p ≥1).

This then implies the existence of exotic baryons (withconfigurations qp+3 qp, p ≥1), and one is led to imagine a tower of exotic hadronic stateswith higher and higher masses.In this letter we first present a more refined analysis of the existing data and then examinemore precisely the role a hadron-scale string theory might play in predicting the densitiesof baryon and meson states. In particular, while the result of Kutasov and Seiberg can beexpected to hold in the asymptotic region (mass M →∞), we find that for energies in theGeV range a na¨ıve hadron-scale string picture implies that the ratio between the numbersof baryon and meson states should in fact oscillate around unity, with mesons favored first,then baryons, then mesons again.The amplitude of this oscillation falls to zero as themass increases (in accordance with the Kutasov-Seiberg result), but we find that for massesbelow 2 GeV, the oscillation is still within its first cycle and can thus accommodate both2

the apparent surplus of lower-energy mesons as well as the surplus of higher-energy baryons.While there is therefore no apparent need for exotic mesons in the mass range Freund andRosner had in mind (1.7 ≤M ≤2 GeV), this oscillating ratio suggests an entirely differentscenario for exotic hadrons: each repeating cycle of the oscillation may correspond to thethreshold for the next-order exotic mesons and baryons. Other scenarios (e.g., involvingglueballs and hybrid quark/gluon states) are possible as well.Let us now be more specific, and first outline some of the basic results of string theory(including that of Kutasov and Seiberg) which will be relevant for our discussion.

Strings areone-dimensional extended objects whose different vibrational and rotational configurationscorrespond to different spacetime particles or states; in general the mass of such a state isgiven bym =r nα′ ,n ∈Z(1)where α′ is a constant characterizing the energy scale of the theory and where n is relatedto the number of vibrational mode-excitations necessary for producing the state. Since theLorentz spin J of such a state must satisfy J ≤n + α0 where α0 is a constant, we have thegeneral resultJ ≤α′ m2 + α0(2)which identifies the constant α′ as the traditional Regge slope.

If the particular string theorycontains both bosonic and fermionic states, we may denote their numbers at each level n asBn and Fn respectively; note that these are the numbers of states or field-theoretic degreesof freedom, and not the number of particles (e.g., spin or isospin multiplets).Anotherwell-known prediction of string theory, then, is the asymptotic exponential growth of thesenumbers as functions of n:Bn, Fn ∼a n−b ec√nas n →∞(3)where the positive constants a, b, and c are theory-specific parameters. Eqs.

(2) and (3)apply in general to all string-type theories. More recently, however, Kutasov and Seiberg3

have obtained a result [3] which applies to those string theories (or more generally, to thosetwo-dimensional conformal field theories) which are free of physical tachyons and whichhave modular-invariant one-loop (toroidal) partition functions.Specifically, if we defineB(N) ≡PNn=0 Bn and F(N) ≡PNn=0 Fn, then Kutasov and Seiberg claim thatlimN→∞[B(N) −F(N)] = 0 ,(4)which in turn implies the weaker constraintlimN→∞[F(N)/B(N)] = 1. (5)We shall require only this weaker form of the Kutasov-Seiberg result; indeed, the strongerversion in Eq.

(4) may not be entirely correct. [5]The extent to which such a string theory can be taken as a theory of hadrons is far fromclear, and therefore in this letter we shall confine ourselves to only those issues which followfrom direct comparisons with the above generic results.

Specifically, we shall assume [1]that one can model hadronic physics as a GeV-scale string theory giving rise to Eqs. (2),(3), and (5), with bosonic states identified as meson degrees of freedom and fermionic statesas baryon degrees of freedom; furthermore, we shall consider only those generic aspects ofstring theory which affect the relative numbers of these states (i.e., their ratio) or theirseparate patterns of growth.

Any other features, such as the specific absolute sizes of B(N)and F(N) or the mapping between particular string configurations and particular hadronicstates, are likely to be highly model-dependent.We have computed the numbers and densities of experimentally-observed meson andbaryon states as functions of their masses. We have included those states containing onlythe three light quarks (u, d, s), both for reasons of experimental statistics [1] and more funda-mentally because hadrons composed of heavy quarks do not lie on linear Regge trajectoriesas a string picture would dictate [Eq.

(2)]. We differ from Ref.

1, however, in recognizingthat although states in string theory are typically of zero width, most of the hadronic statesor resonances are quite broad. Therefore, we have taken the hadronic density of states tobe a sum of normalized Breit-Wigner distributions:4

dNdm =12πXiWiΓi(m −Mi)2 + Γi2/4(6)where Mi and Γi are respectively the masses and widths of the observed states, [6] and whereWi are their multiplicities [i.e., the number of states per resonance, or (2I + 1)(2J + 1) fora charge self-conjugate state of spin J and isospin I, and twice that otherwise]. In Fig.

1we have plotted the total hadronic density of states as a function of m, and it is clearthat this density experiences the exponential (Hagedorn-like) growth suggested in Eq. (3)with Hagedorn temperature [2] TH ≡(c√α′)−1 ≈250 MeV, at least for masses up to 2GeV.

Barring unexpected physics, the failure of the curve in Fig. 1 to maintain this growthbeyond 2 GeV is likely to be a reflection of current experimental limitations.

Thus, we shallhenceforth limit our attention to the experimental data below 2 GeV.In Fig. 2 we have plotted the separate numbers (or integrated densities) of baryon andmeson states with masses m ≤M as functions of M. In order to facilitate a comparison withEq.

(5), we have also plotted their ratio as the shaded region in Fig. 3: this shaded regionindicates the uncertainty in the ratio function due to the hadronic widths, with the upperborder of the region corresponding to the Breit-Wigner densities in Eq.

(6) and the lowerborder corresponding to the zero-width case. Either way, several features are immediatelyapparent, among them the pronounced surplus of mesons below 1.5 GeV and the pronouncedsurplus of baryons above this energy; indeed, this ratio shows no sign of a plateau near unity.This figure thus clearly indicates that it is hardly compelling to interpret this mass regionas the region of onset of Kutasov-Seiberg asymptotic behavior.

It is in fact straightforwardto estimate the string-level n in Eq. (1) to which a mass of 1.5 GeV corresponds: taking themeasured value of the hadronic Regge slope α′ ≈0.9 (GeV)−2, we obtain n ≈2.

Indeed,the entire regions < 2 GeV correspond only to string-levels n ≤4. Thus, even though theselow-lying levels experience the asymptotic growth in Eq.

(3), they clearly need not manifestthe asymptotic behavior predicted in Eq. (5); indeed, the latter asymptotic behavior occursonly at higher energies.Therefore, in order to determine the characteristics of the approach towards asymptotic5

behavior, we have calculated the ratio functions R(N) = F(N)/B(N) predicted by a varietyof different string theories (or string “models”) of the sort to which Eq. (5) should apply.While certain features of this function vary greatly and are highly model-dependent, others– such as the exponential increase in the level degeneracies [Eq.

(3)] or the existence ofa Kutasov-Seiberg limit [Eq. (5)] – indeed appear to be generic.In particular, we findan important third universal feature: [5] as N increases, we find that the function R(N)oscillates around unity, with the amplitude of this oscillation decreasing with increasingN.This “damped” oscillation, periodic in n = α′M2, is of course consistent with theKutasov-Seiberg result in Eq.

(5). Such an oscillation between bosonic and fermionic statesis a consequence (and in fact the signature) of an underlying string symmetry known asmodular invariance, and the wavelength λ of this oscillation is determined only by theenergy scale of the theory, [5] λ = 4/α′.

The amplitude, on the other hand, is somewhatmodel-dependent, and in fact vanishes in the case of supersymmetry: indeed, the only wayto break supersymmetry while preserving modular invariance is to do so in this regularoscillatory manner. [5] In Fig.

3 we have superimposed the results of a calculation based ona typical non-supersymmetric string model, plotting R(N) vs. M ≡qN/α′.In the mass range M ≤2 GeV, the behavior of the string ratio in Fig. 3 is certainlyconsistent with the observed ratio: this oscillation typically begins with R < 1 (at N = 0),first crosses R = 1 at N = 2 (corresponding to M ≈1.5 GeV), and then increases beyond 1as M approaches 2 GeV.

Thus we see that the sign of the oscillation, as well as the positionof the first node, are consistent with the data, and a surplus of mesons below 1.5 GeV aswell as a surplus of baryons above 1.5 GeV are easily accommodated. Thus, on the basis ofa comparison between these two figures in the M ≤2 GeV range, we find that we need notclaim a deficit of meson states with masses just above 1.5 GeV.It will be interesting, however, to see whether the entire string-theoretic oscillation is ul-timately realized at higher energies.

While such an oscillation between bosonic and fermionicstates has not been observed experimentally, we have seen in Fig. 1 that many hadronic stateswith energies above 2 GeV must be missing if Hagedorn-like growth is to be maintained in6

that region. That many such states are missing is also expected from an SU(3) pictureas well as from conventional Regge-trajectory arguments.

Such an oscillation, therefore,remains entirely possible.It is important to bear in mind that we have focused on only the generic features pre-dicted by a generic string-type theory, and one would need to further refine a particularstring picture in order to expect a more quantitative agreement between the observed andpredicted ratio functions. For example, the string theories we have examined here are intrin-sically non-interacting: all of their states (or particles) have zero width, and can populateonly the discrete energy levels indicated in Eq.

(1). This is the origin of the sharp changesin the string ratio function in Fig.

3, and a more fully-developed string theory incorporatingparticle interactions would undoubtedly yield a smoother, more continuous ratio function.Furthermore, dynamical considerations are also at the root of the relatively small size ofthe experimentally observed ratio function at masses M ≤1 GeV: the lowest-lying mesons(i.e., the pions) have masses protected by a nearly-unbroken chiral symmetry, while themasses of the lowest-lying baryons (i.e., the proton and neutron) are entirely unprotectedand consequently much greater. This is in contrast to non-interacting string theories, whichgenerically contain both bosons and fermions at the (exactly) massless level.

A fully inter-acting string theory, therefore, should be expected to yield a closer agreement between theratio functions, especially in the lower-mass region. On the other hand, the oscillations inthe ratio function are of a more universal nature, and although interactions can be expectedto make them smooth, they should remain quite pronounced in the region M < 4 GeVwhere their amplitudes are large.Given that string theories generically lead to such oscillations, and given that we cannotsoon expect to observe all existing states in the several-GeV region, it is natural to try to pre-dict how these oscillations might arise within the context of a more traditional quark/gluonpicture.

While the string theories themselves unambiguously predict which string vibra-tional/rotational configurations are ultimately responsible for producing these oscillations,[5] one must specify or choose a particular mapping between these configurations and the7

various quark/gluon states in order to interpret these oscillations in terms of selected groupsof baryons and mesons. The results are then highly model-dependent.

Therefore, ratherthan advocate a particular string-to-hadron mapping, we will simply propose two possibleresulting scenarios which naturally extend the ideas of Ref. 1.One natural scheme which might lead to such a regular, periodic meson/baryon oscillationinvolves exotic hadrons – i.e., mesons with quark structure (qq)p+1 and baryons with quarkstructure qp+3 qp for p ≥1.The special cases with p = 0 of course correspond to theordinary mesons and baryons which respectively dominate the two halves of the first cycleof the oscillation.

It is thus natural to speculate that such a repeating pattern of oscillationsis the result of regularly-spaced thresholds for the pth exotic hadrons, implying alternatingmass regions in which either the pth exotic mesons or baryons dominate:(qq)p+1 mesons :(p + 1/4) λ ≤M2 ≤(p + 1/2) λqp+3 qp baryons :(p + 3/4) λ ≤M2 ≤(p + 1) λ(7)where λ = 4/α′ ≈4.4 (GeV)2. Such an ordering of thresholds is in fact consistent with alter-native analyses.

[1,7] Another scenario involves not only glueballs but hadron/glue “hybrids”,for such states –if color-neutral– are in principle also present in a quark-gluon theory. Whileglueballs are necessarily bosonic, hybrid states can contribute to both bosonic and fermionicdegrees of freedom depending on their quark content.

In this scenario, then, each subsequentcycle of our oscillation corresponds to the crossing of the threshold for the next-order hybridhadrons (i.e., hadrons with one additional gluonic insertion), with the wavelength λ = 4/α′of our oscillation representing the mass shift resulting from such gluonic insertions. Thus,this picture too can naturally explain the regularity of the string-predicted oscillation.

Note,however, that any such picture necessarily implies the existence of exponentially increasingnumbers of fundamentally new hadronic states at each of the mass regions listed in Eq. (7)– starting with, in particular, several hundred between 2 and 2.3 GeV.In summary, then, we find that a generic hadron-scale string theory is consistent withthe observed ratio of baryon and meson states; in particular, agreement with string theory8

does not require the existence of “missing mesons” (ordinary or exotic) in the mass regionjust above 1.5 GeV. On the other hand, we find that string theory and modular invariancepredict a fermion/boson ratio which oscillates around unity as the mass increases, with theamplitude of these oscillations steadily decreasing.

Such a picture therefore lends itself toa variety of interpretations involving exotic and/or hybrid hadrons, with each cycle of thisoscillation corresponding to the thresholds for the next-order mesons and baryons. It will beinteresting to see whether such pictures can be realized in more traditional (e.g., statisticalor potential) quark-models as well.ACKNOWLEDGMENTSWe are pleased to thank our colleagues at McGill, especially C.S.

Lam and B. Margolis,for many fruitful discussions. This work was supported in part by NSERC (Canada) andles fonds FCAR (Qu´ebec).9

REFERENCESREFERENCES(a) Electronic mail address: cudell@hep.physics.mcgill.ca(b) Electronic mail address: dien@hep.physics.mcgill.ca1P.G.O. Freund and J.L.

Rosner, Phys. Rev.

Lett. 68, 765 (1992).2R.

Hagedorn, Nuovo Cimento 56A, 1027 (1968).3D. Kutasov and N. Seiberg, Nucl.

Phys. B358, 600 (1991).4S.

Godfrey and N. Isgur, Phys. Rev.

D32, 189 (1985); S. Capstick and N. Isgur, Phys. Rev.D34, 2809 (1986).5K.R.

Dienes, McGill University preprint McGill/92-08 (to appear).6Particle Data Group, J.J. Hern´andez et. al., Phys.

Lett. B239, 1 (1990).7J.R.

Cudell, K.R. Dienes, and B. Margolis, work in progress.10

11

FIGURESFIG. 1.

Total density of observed hadronic states as function of mass, along with best-fit toHagedorn form of Ref. 2.12

FIG. 2.

Total numbers of observed baryons (solid line) and mesons (dashed line) with masses≤M, as functions of M.13

FIG. 3.

Shaded region: observed ratio of numbers of baryon and mesons, as discussed in text.Solid line: ratio function from a typical string model.14

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