aa r X i v : . [ h e p - ph ] O c t version 28.9.2009 Hadron mass generation and the strong interaction
H.P. MorschInstitute for Nuclear Studies, Pl-00681 Warsaw, Poland ∗ Abstract
Based on a Lagrangian with a coupling of two gluons to J π = 0 + (the quan-tum numbers of the vacuum) which decay to q ¯ q pairs, a model is presented, inwhich hadrons couple directly to the absolute vacuum of fluctuating gluon fields.By self-consistency requirements the confinement potential as well as q ¯ q densitiesand masses are obtained, which are in good agreement with experimental data onscalar and vector mesons. In comparison with potential models additional statesare predicted, which can explain the large continuum of scalar mesons in the lowmass spectrum and new states detected recently in the charm region.The presented model is consistent with the concept, that the hadron masses can beunderstood by binding effects of the quarks.PACS/ keywords: 12.38.Aw, 12.38.Lg, 12.39.Mk/ Gluon-gluon coupling to q ¯ q -pairs, generation of stable q ¯ q -systems assuming massless quarks, confinement po-tential, masses of 0 ++ and 1 −− mesons. In the hierarchy of quantum systems hadrons represent the smallest complex substructuresknown inside of atoms and nuclei. This is supported by the property of asymptoticfreedom [1] of the strong interaction. Therefore, hadrons can be related directly to theabsolute vacuum of fluctuating gluon fields (with average energy ¯ E vac = 0) if the quarkmasses are zero. In quantum chromodynamics (QCD) the vacuum is more complex witha finite mass of the quarks, which may be coupled to scalar Higgs particles, subject ofextensive searches with the available and planned high energy experiments at Fermilaband CERN.To investigate hadron mass generation we start from a Lagrangian, in which two gluonscouple to scalar fields, from which q ¯ q pairs are emitted. The possibility that two gluonscoupled to 0 + may produce colour singlet bound states has been mentioned already byCornwall [2], but in context to the structure of QCD. Since the overlap of gluon fields is1on-local, such a model could serve also as an effective theory to investigate the strongscalar fields observed in hadron excitations [3] and scattering [4]–[7], which may be difficultto extract from QCD.Assuming a scalar coupling of gluon fields of the form gg → ( q ¯ q ) n we write the Lagrangianin the form L SI = ¯Ψ iγ µ D µ Ψ −
14 ( F µν F µν − G µ G µ ) , (1) with D µ being the covariant derivative D µ = ∂ µ − ig s A µ and F µν the Abelian field strengthtensor F µν = ∂ µ A ν − ∂ ν A µ , with A µ being the gluon fields. G µ G µ couples two gluon fieldswith J π = 0 + to q ¯ q -pairs with G µ = − g s A µ [ b ( ¯Ψ V g Ψ) + b ( ¯Ψ V g Ψ) + b ( ¯Ψ V g Ψ) + ... ] , (2)where ( ¯Ψ V g Ψ) stands for the creation of q ¯ q -pairs, which interact by gluon exchange.Corresponding Feynman diagrams are shown in the upper part of fig. 1.The structure of L s = G µ G µ implies a colour neutral coupling of two gluon fields. Hence,the symmetry is simply isospin SU(2) (two quarks with different charge and one gluon)without colour and (because of massless quarks as seen below) without flavour degree offreedom. By the coupling of two gluons to J π = 0 + the Lagrangian has no chiral symmetry,leading naturally to the sequence of hadronic states as observed experimentally.A two-gluon field is produced only, if there is spacial overlap of two gluon fields. We writethe radial part of L s by a matrix elementΦ( r − r ) = α s < A ( r ) | [ b ( a † q V g a q ) + b ( a † q V g a q ) + ... ] | A ( r ) > , (3)where A i ( r ) are radial gluon fields, a † q and a q quark and antiquark creation operators,and V g an interaction dominated by 1-gluon exchange between these quarks . For thefollowing discussion only the first term in eq. (3) is needed, contributions due to ( q ¯ q ) contributions will be discussed briefly at the end of the paper.Due to the non-local 2-gluon field, given by a 2-gluon density ρ Φ ( r ), the recoiling localquark fields are also smeared out and a † q V g a q can be written as a q-q potential V qq ( r )described by folding a q ¯ q -density with the gluon-exchange interaction. The fact, that More-gluon exchange as well as spin-spin and spin-orbit effects have not been considered. qq ( r ) is a scalar potential but the created q ¯ q -pair has negative parity, requires a p-wave q ¯ q -density ρ pq ¯ q ( ~r ) = ρ q ¯ q ( r ) Y ,m ( θ, φ ) leading to V qq ( r ) = Z dr ′ ρ pq ¯ q (¯ r ′ ) Y ,m ( θ ′ , φ ′ ) V g ( r − r ′ ) , (4)where V g ( r ) can act only within the density ρ Φ ( r ). Therefore the interaction has to becut towards large radii and we use the following form V g ( r ) = ( − α s /r ) e − cr (5)with cut-off parameter c determined in a self-consistent fit of the 2-gluon density. Note,that eq. (3) indicates a departure from a purely relativistic description with a lifetime ofthe created system ∆ t . Causality is fulfilled, if ∆ t > ( r − r ′ ) /c . The shape of the foldingpotential (4) has been calculated by multiplying the density and potential in momentumspace and retransforming the product to r-space.The p-wave character of the q ¯ q -density gives rise to the constraint < r q ¯ q > = R dτ rρ q ¯ q ( r ) =0 and thus ρ q ¯ q ( r ) = √ β · d/dr ) ρ Φ ( r ) , (6)where β is determined from the condition < r q ¯ q > = 0. A consequence of eq. (3) is, thatthe radial dependence of V qq ( r ) should be the same as ρ Φ ( r ) ρ Φ ( r ) = V qq ( r ) . (7)From the different relations between densities and potential in eqs. (4), (6) and (7) thetwo-gluon density is determined. Self-consistent solutions of eq. (7) are obtained assuminga form ρ Φ ( r ) = ρ o [ exp {− ( r/a ) κ } ] with κ ∼ . c , which yields a mean square radius of the effectiveinteraction between 20 and 80 % larger than < r > . The self-consistency condition israther strict (see the lower parts of fig. 1): using a pure exponential form ( κ =1) a verysteep rise of ρ Φ ( r ) is obtained for r → κ =2) no consistency is obtained:normalised to the inner part of ρ Φ ( r ), the deduced potential falls off more rapidly towardslarger radii than the density. Only by a density with κ ∼ . > < gg → q ¯ q in question is elasticand consequently the created q ¯ q -pair has no mass. However, if we take a finite mass ofthe created quarks of 1.4 GeV (such a mass has been assumed in potential models [8]for systems of similar size), the dashed line in the lower part of fig. 2 is obtained andno self-consistent solution is possible. Thus, our solutions require massless quarks andconsequently the deduced hadronic systems can be related directly to the absolute vacuumof fluctuating gluon fields.For a quark-gluon system with finite mean square radius as shown in fig. 2 the corre-sponding binding potential can be obtained from a three-dimensional reduction of theBethe-Salpeter equation in form of a (relativistic) Schr¨odinger equation − (cid:18) ¯ h µ Φ h d dr + 2 r ddr i − V Φ ( r ) (cid:19) ψ Φ ( r ) = E i ψ Φ ( r ) , (9)where µ Φ is the mass parameter, which is related to the mass m Φ of the q ¯ q system by µ Φ = m Φ − δm with a correction δm = 0 only for very light systems. Further, ψ Φ ( r ) isthe wave function, which is given for a system of uncorrelated gluons by | ψ Φ ( r ) | = ρ Φ ( r ).The dependence of the potential as a function of ψ Φ ( r ) is then given by V Φ ( r ) = ¯ h µ Φ (cid:16) d ψ Φ ( r ) dr + 2 r dψ Φ ( r ) dr (cid:17) ψ Φ ( r ) + V o . (10)Inserting the form of the density in eq. (8) yields explicitely V Φ ( r ) = ¯ h µ Φ h κa ( ra ) κ − [ κ ( ra ) κ − ( κ + 1)] i + V o . (11)Since the (2 g − q ¯ q ) system couples to the vacuum, V o is assumed to be zero. Withthe values of κ and a from the self-consistent solutions (8) and µ Φ ∼ V conf = − α s /r + br deduced from potential models [8] and the latticedata of Bali et al. [9]. Further, the potential (11) is the same for other solutions of smaller4r larger radii and can therefore be identified with the ’universal’ confinement potential. Itis important to note, that the self-induced confinement potential (11) reproduces the 1/r +linear form without any assumption on its distance behavior; this is entirely a consequenceof the deduced radial form (8) of the two-gluon density. Interestingly, because of the rathersimple confinement mechanism in our approach, a direct connection to strings (which havebeen related to the linear part of confinement potential) appears possible.To determine bound state energies of basic scalar and vector q ¯ q states (with J P C =0 ++ and 1 −− ) binding in the self-induced confinement potential (11) but also in the q-qpotential has to be considered, which (different from the confinement potential) dependsstrongly on the radius of the density. For smaller radii a deepening of this potential isobserved, which leads to more strongly bound states.We define the mass of the system by the energy to balance binding (this is also the wayhadron masses are observed). This yields M i = − E qq + E i , (12)where E qq and E i are the binding energies in V qq ( r ) and V Φ ( r ), respectively (note that theeigenstates of V qq ( r ) have negative energy).We apply another constraint (generation of mass by binding), requiring that the mass M o of the lowest bound state is equal to m Φ (appearing in the mass parameter µ Φ = m Φ − δm with δm ∼ m Φ ∼ < r > . for scalar states with acoupling constant α s decreasing for smaller systems. For the five systems in table 1 wefind values of α s of about 1.3, 0.5, 0.4, 0.32 and 0.23. Together with other solutions wefind α s ( M o ) ≈ . α QCDs ( Q ) for M o = Q up to large masses (with values of α QCDs ( Q )from the systematics in ref. [10]). This yields evidence for asymptotic freedom also in ourmodel. More details will be discussed elswhere.So far we have no constraint on the radial extent of the (2 g − q ¯ q ) system, giving riseto continuous mass spectra. Discretisation is provided by a vacuum potential sum ruleΦ vac ( r ) = P n Φ n ( r ), requiring that the sum of two-gluon matrix elements (3) (whereΦ n ( r ) = α ns ρ Φ n ( r )) is equal to the total 1-gluon exchange force in the vacuum. For thiswe take the simplest form, demanding that the total cut-off by the finite size of theoverlapping gluon fields shows the same 1 /r dependence as the 1-gluon exchange force.5able 1: Deduced masses (in GeV) of scalar and vector q ¯ q states in comparison withknown 0 ++ and 1 −− mesons [10]. Only the 1s states in the q-q potential are given. Theradii of the known states of mass M o are fine-tuned to agree with experimental data.Solution (meson) M M M M exp M exp M exp ++ σ ± ± ± ++ f o ± ± ± −− ω ± ± ± ++ f o ± ± ± −− Φ 1.00 1.69 2.27 1.02 1.68 ± ++ not seen 4.5 ± o +0.5 M o +1.0 —1 −− J/ Ψ 3.10 3.68 4.19 3.097 3.686 (4.160)5 0 ++ not seen 22.0 ± o +0.4 M o +0.7 —1 −− Υ 9.46 9.99 10.36 9.46 10.023 10.355From this we obtain Φ vac ( r ) = − ˜ α s r = − X n α ns ρ Φ n ( r ) . (13)Five self-consistent solutions for scalar states below 50 GeV have been extracted withmean square radii < r > of about 0.49, 0.26, 0.12, 0.06, and 0.02 fm , respectively,and masses M o given in table 1. In the lower part of fig. 3 the different solutions Φ n ( r )for n=1-5 are shown together with their sum (solid line). This is compared to the sumrule (13) with ˜ α s =0.35 (lower dot-dashed line). Good agreement is obtained, indicatingthat the sum rule is fulfilled. Only for radii < q ¯ q density has to be replaced by the corresponding s-wave density.This leads to momentum distributions of the vector states shifted to smaller momenta(and corresponding smaller masses) than the scalar states. We obtain mean square radii < r > of about 0.38, 0.24, 0.10, and 0.04 fm and masses given in table 1, which arein good agreement with the masses of the strong 1 −− mesons (together with their radialexcitations) of the (isoscalar) “flavour families” ω , Φ, J/ Ψ and Υ. The masses of the6nown 0 ++ states are also in general agreement with experiment: the lowest 0 ++ statecorresponds to the σ (600) meson, which has been clearly identified [11] as a broad mesonresonance in J/ Ψ-decay. The next 0 ++ states at 1.3 and 2.3 GeV may correspond to thescalar resonances f o (1300), see also ref. [12], and f o (2300), whereas the higher 0 ++ states(not found so far) may not be observable in e + - e − collision experiments.As compared to potential models with finite quark masses, as e.g. in ref. [8], we obtainsignificantly more states, bound states in the confinement and in the q-q potential. Thesolutions in table 1 correspond only to the 1s levels in the q-q potential, in addition we havecalculated Ns levels for N=2, 3, and 4. Most of these states, however, have a relativelysmall mass below 3 GeV. As the q-q potential is Coulomb like, it creates a continuumof Ns levels which ranges down in mass to the threshold region. This continuum shouldmix with the states in the confinement potential, giving rise to large scalar phase shifts atlow energies, which are observed experimentally but so far not well understood in othermodels.Concerning masses above 3 GeV, solution 5 of table 1 yields additional 0 ++
2s and 3sstates in the q-q potential at masses of about 12 and 8.8 GeV, respectively, whereas anextra 1 −−
2s state is obtained (between the most likely Ψ(3s) and Ψ(4s) states at 4.160GeV and 4.415 GeV) at a mass of about 4.2 GeV. This state may be identified with therecently discovered X(4260), see ref. [10]. Corresponding excited states in the confinementpotential (11) should be found at masses of 4.9, 5.3 and 5.5 GeV with uncertainties of0.2-0.3 GeV.By identifying the deduced q ¯ q states with known mesons we can check the overall con-sistency of our model by the observed widths of these states. For the lifetime we assume < ∆ t > = < r > / /c and get for the heavier mesons in table 1 values which are smallcompared to the lifetime deduced from the relatively small experimental widths. Dif-ferently, for the lightest meson in table 1 we obtain a value of < ∆ t > of 2.4 10 − scorresponding to a width of about 300 MeV, which is less than the width ( ∼ σ (600). However, as the Coulomb like q-q potential gives rise toa low energy continuum of Ns states, the width of the lowest bound state in table 1 hasto be much smaller, indicating that also in this case our approach is valid.Finally, possible ( q ¯ q ) contributions corresponding to the second term in eq. (3) are ad-7ressed. In the results shown in fig. 2 and the upper part of fig. 4 the folding potentialdeviates from ρ Φ ( r ) only at small radii. This difference could be filled by a small ( q ¯ q ) component. This is supported by Monte Carlo simulations [13], in which good agreementbetween ρ Φ ( r ) and V qq ( r ) in eq. (7) is obtained in the entire r and Q region by the inclusionof a ( q ¯ q ) contibution.In conclusion, a model has been presented, in which hadron masses are described asbound states of quarks with a very simple structure of the vacuum. This leads to a gooddescription of the confinement potential and hadron masses. Other results, including adiscussion of asymptotic freedom and the stability of baryons will be discussed later.We thank P. Decowski, M. Dillig (deceased), A. Kupsc and P. Raczka among many othercollegues for fruitful discussions, valuable comments and the help in formal derivations.Special thanks to P. Zupranski for writing the Monte Carlo code, for numerous conversa-tions and clarificatons. ∗ postal address: Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, D-52425 J¨ulich, Ger-many, E-mail: [email protected] 8 eferences [1] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H.D. Politzer, Phys.Rev. Lett. 30, 1346 (1973); see also the comments in D.J. Gross, Nobel lecture (2004)[2] Cornwall, Phys. Rev. D 26, 1453 (1982)[3] H.P. Morsch and P. Zupranski, Phys. Rev. C 71, 065203 (2005) and refs. therein,H.P. Morsch, Z. Phys. A 350, 61 (1994)[4] R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189; and refs. therein[5] H.P. Morsch, W. Spang and P. Decowski, Phys. Rev. C 67, 064001 (2003)[6] A. Donnachie and P.V. Landshoff, Nucl. Phys. B 231 (1984) 189 and B 311 (1989)509; P.V. Landshoff and O. Nachtmann, Z. Phys. C 35, 405 (1987); and refs. therein[7] F. Halzen, G. Krein and A.A. Natale, Phys. Rev. D 47, 295 (1993);M.G. Gay Ducati, F. Halzen and A.A. Natale, Phys. Rev. D 48, 2324 (1993);and refs. therein[8] R. Barbieri, R. K¨ogerler, Z. Kunszt, and R. Gatto, Nucl. Phys. B 105, 125 (1976);E. Eichten, K.Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, Phys. Rev. D 17,3090 (1978); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985); D. Ebert, R.N.Faustov, and V.O. Galkin, Phys. Rev. D 67, 014027 (2003); and refs. therein[9] G.S. Bali, K. Schilling, and A. Wachter, Phys. Rev. D 56, 2566 (1997)[10] Review of particle properties, C. Amsler et al., Phys. Lett B 667, 1 (2008); andhttp://pdg.lbl.gov/[11] M. Ablikim, et al., hep-ex/0406038 (2004); see also D.V. Bugg, hep-ex/0510014(2005) and refs. therein[12] V.V. Anisovitsch and A.V. Sarantsev, Eur. Phys. J. A 16, 229 (2003) Phys. Rev.Lett. 95, 142001 (2005)[13] Monte Carlo simulations of 2 g → ( q ¯ q ) + ( q ¯ q ) using the q ¯ q potential in eq. (4) andquark momenta determined randomly, H.P. Morsch and P. Zupranski, to be published9 Figure 1: Upper part: Relevant Feynman diagrams 2 g → q ¯ q and 2 g → ( q ¯ q ) (q denotesquark or antiquark). Lower parts: density ρ Φ ( r ) with < r > =0.06 fm (dot-dashed lines)and folding potential (4) (solid lines) for κ =1, 1.5 and 2, respectively.10 Figure 2: Upper part: Self-consistent solution of eq. (4) for a scalar state with two-gluondensity and q-q potential given by dot-dashed and solid lines, respectively. Lower part:Same as in the upper part but transformed to Q -space (multiplied by Q ). The dashedline corresponds to a calculation assuming quark masses of 1.4 GeV.11 Figure 3: Upper part: Confinement potential from lattice calculations [9] in comparisonwith the qq
2s state is obtained (between the most likely Ψ(3s) and Ψ(4s) states at 4.160GeV and 4.415 GeV) at a mass of about 4.2 GeV. This state may be identified with therecently discovered X(4260), see ref. [10]. Corresponding excited states in the confinementpotential (11) should be found at masses of 4.9, 5.3 and 5.5 GeV with uncertainties of0.2-0.3 GeV.By identifying the deduced q ¯ q states with known mesons we can check the overall con-sistency of our model by the observed widths of these states. For the lifetime we assume < ∆ t > = < r > / /c and get for the heavier mesons in table 1 values which are smallcompared to the lifetime deduced from the relatively small experimental widths. Dif-ferently, for the lightest meson in table 1 we obtain a value of < ∆ t > of 2.4 10 − scorresponding to a width of about 300 MeV, which is less than the width ( ∼ σ (600). However, as the Coulomb like q-q potential gives rise toa low energy continuum of Ns states, the width of the lowest bound state in table 1 hasto be much smaller, indicating that also in this case our approach is valid.Finally, possible ( q ¯ q ) contributions corresponding to the second term in eq. (3) are ad-7ressed. In the results shown in fig. 2 and the upper part of fig. 4 the folding potentialdeviates from ρ Φ ( r ) only at small radii. This difference could be filled by a small ( q ¯ q ) component. This is supported by Monte Carlo simulations [13], in which good agreementbetween ρ Φ ( r ) and V qq ( r ) in eq. (7) is obtained in the entire r and Q region by the inclusionof a ( q ¯ q ) contibution.In conclusion, a model has been presented, in which hadron masses are described asbound states of quarks with a very simple structure of the vacuum. This leads to a gooddescription of the confinement potential and hadron masses. Other results, including adiscussion of asymptotic freedom and the stability of baryons will be discussed later.We thank P. Decowski, M. Dillig (deceased), A. Kupsc and P. Raczka among many othercollegues for fruitful discussions, valuable comments and the help in formal derivations.Special thanks to P. Zupranski for writing the Monte Carlo code, for numerous conversa-tions and clarificatons. ∗ postal address: Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, D-52425 J¨ulich, Ger-many, E-mail: [email protected] 8 eferences [1] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H.D. Politzer, Phys.Rev. Lett. 30, 1346 (1973); see also the comments in D.J. Gross, Nobel lecture (2004)[2] Cornwall, Phys. Rev. D 26, 1453 (1982)[3] H.P. Morsch and P. Zupranski, Phys. Rev. C 71, 065203 (2005) and refs. therein,H.P. Morsch, Z. Phys. A 350, 61 (1994)[4] R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189; and refs. therein[5] H.P. Morsch, W. Spang and P. Decowski, Phys. Rev. C 67, 064001 (2003)[6] A. Donnachie and P.V. Landshoff, Nucl. Phys. B 231 (1984) 189 and B 311 (1989)509; P.V. Landshoff and O. Nachtmann, Z. Phys. C 35, 405 (1987); and refs. therein[7] F. Halzen, G. Krein and A.A. Natale, Phys. Rev. D 47, 295 (1993);M.G. Gay Ducati, F. Halzen and A.A. Natale, Phys. Rev. D 48, 2324 (1993);and refs. therein[8] R. Barbieri, R. K¨ogerler, Z. Kunszt, and R. Gatto, Nucl. Phys. B 105, 125 (1976);E. Eichten, K.Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, Phys. Rev. D 17,3090 (1978); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985); D. Ebert, R.N.Faustov, and V.O. Galkin, Phys. Rev. D 67, 014027 (2003); and refs. therein[9] G.S. Bali, K. Schilling, and A. Wachter, Phys. Rev. D 56, 2566 (1997)[10] Review of particle properties, C. Amsler et al., Phys. Lett B 667, 1 (2008); andhttp://pdg.lbl.gov/[11] M. Ablikim, et al., hep-ex/0406038 (2004); see also D.V. Bugg, hep-ex/0510014(2005) and refs. therein[12] V.V. Anisovitsch and A.V. Sarantsev, Eur. Phys. J. A 16, 229 (2003) Phys. Rev.Lett. 95, 142001 (2005)[13] Monte Carlo simulations of 2 g → ( q ¯ q ) + ( q ¯ q ) using the q ¯ q potential in eq. (4) andquark momenta determined randomly, H.P. Morsch and P. Zupranski, to be published9 Figure 1: Upper part: Relevant Feynman diagrams 2 g → q ¯ q and 2 g → ( q ¯ q ) (q denotesquark or antiquark). Lower parts: density ρ Φ ( r ) with < r > =0.06 fm (dot-dashed lines)and folding potential (4) (solid lines) for κ =1, 1.5 and 2, respectively.10 Figure 2: Upper part: Self-consistent solution of eq. (4) for a scalar state with two-gluondensity and q-q potential given by dot-dashed and solid lines, respectively. Lower part:Same as in the upper part but transformed to Q -space (multiplied by Q ). The dashedline corresponds to a calculation assuming quark masses of 1.4 GeV.11 Figure 3: Upper part: Confinement potential from lattice calculations [9] in comparisonwith the qq ¯ qq