We argue that three-quark excited states naturally group into quartets, split
into two parity doublets, and that the mass splittings between these parity
partners decrease higher up in the baryon spectrum. This decreasing mass
difference can be used to probe the running quark mass in the mid-infrared
power-law regime. A measurement of masses of high-partial wave Delta*
resonances should be sufficient to unambiguously establish the approximate
degeneracy. We test this concept with the first computation of excited high-j
baryon masses in a chirally invariant quark model.Comment: 4 pages, 4 figures. submitted to Phys Rev Letter

Bicudo, P.

Cardoso, M.

Llanes-Estrada, Felipe J.

Van Cauteren, T.

English

arXiv

P. Bicudo

M. Cardoso

T. Van Cauteren

Felipe J. Llanes-Estrada

Crossref

Probing the Infrared Quark Mass from Highly Excited Baryons

We argue that three-quark excited states naturally group into quartets, split into two parity doublets, and that the mass splittings between these parity partners decrease higher up in the baryon spectrum. This decreasing mass difference can be used to probe the running quark mass in the mid-infrared power-law regime. A measurement of masses of high-partial wave Delta* resonances should be sufficient to unambiguously establish the approximate degeneracy.  We test this concept with the first computation of excited high-j baryon masses in a chirally invariant quark model

Bicudo, Pedro

Cardoso, Marco

Van Cauteren, Tim

Llanes Estrada, Felipe J.

Ghent University Academic Bibliography

arXiv:0902.3613v2  [hep-ph]  4 Aug 2009Probing the infrared quark mass from highly excited baryonsP. Bicudo1, M. Cardoso1, T. Van Cauteren2, Felipe J. Llanes-Estrada31Instituto Superior Te´cnico, Lisboa, Portugal; 2Dept. Sub. and Rad. Phys.,Ghent University, Belgium; 3Dept. F´ısica Teorica I, Univ. Complutense, Madrid, SpainWe argue that three-quark excited states naturally group into quartets, split into two paritydoublets, and that the mass splittings between these parity partners decrease higher up in thebaryon spectrum. This decreasing mass difference can be used to probe the running quark mass inthe mid-infrared power-law regime. A measurement of masses of high-partial wave ∆∗ resonancesshould be sufficient to unambiguously establish the approximate degeneracy. We test this conceptwith the first computation of excited high-j baryon masses in a chirally invariant quark model.Quantum Chromodynamics (QCD) has been thor-oughly tested in high-energy physics, through hadronjets, Drell-Yan processes, e−e+ annihilation, and deepinelastic scattering. Low-energy QCD manifests chiralsymmetry breaking (χSB) that enhances quark masses inthe infrared (IR), generating most of the visible mass inour universe, and that removes the degeneracy from theground states of light hadron spectra (so light hadronsdo not come in parity doublets). The excited light-quark baryon spectrum, with masses above that of the∆-resonance (1232 MeV), has not been accessible to anyform of perturbation theory, effective or fundamental.Insensitivity to chiral symmetry breaking, recentlystressed by Glozman [1, 2, 3], and retrospectively presentin the excited meson spectra computed in chirally invari-ant quark models [4, 5] has led many hadron physicists toaccept that spontaneous χSB, the salient feature of thelow hadron spectrum, actually becomes less importantfor highly excited resonances. For these, chiral symme-try is asymptotically realized in Wigner mode, arranginghadrons in degenerate chiral multiplets [6, 7]. We haverealized that a formalization of this statement is thatthe ratio of the quark mass to its momentum 〈mk〉 pro-vides a new perturbative parameter to study some as-pects of the excited hadron spectrum when the typicalmomentum becomes higher than the running mass. Rea-soning in reverse, we propose employing a cancellationin mass differences of very excited resonances, to accessthis small parameter, and hence the quark mass. Wealso show analytically and numerically that three-quarkstates naturally group into quartets with two states ofeach parity. Diagonalizing the chiral charge in terms ofquarks, the quartet is split into two parity doublets, andall mass splittings tend to decrease when going higher inthe spectrum. For simplicity and to minimize the impactof molecular meson-nucleon configurations [8], we studythe maximum-spin excitations ∆∗ of the Delta baryon.The u or d quark mass [9] (understood as taken fromthe full two-point function of the field theory), is theo-rized to vary fromm ≃ 1−6 MeV at a high, perturbativemomentum k scale (typically≫ ΛQCD ∼ 210 MeV), to aconstituent mass of circa 300 MeV at vanishing momen-tum k ≃ ΛQCD. This IR enhancement of two orders of0 2000 4000k (MeV)0100200300400m(k) (MeV)Constituent  Quark ModelNambu-Jona-LasinioLinear confinement Landau gauge latticeCoulomb gauge latticeDyson-SchwingerFIG. 1: IR enhancement of the light quark mass due to spon-taneous χSB. Shown are quark masses in the main approachesto QCD, all multiplied by an arbitrary factor to match themat quark momentum norm k → 0. (References in the text.)magnitude is shown in Fig. 1. The simplest constituentquark model considers a constant mass [10], withoutthe characteristic step-function of the Nambu and Jona-Lasinio model (also shown). Breaking chiral symmetrywith a linear potential forces a power-law m → Ck4form(k)≪ k. Lattice QCD [11] and Schwinger-Dyson equa-tions [12], both Euclidian approaches to QCD, also showpower-law decays of the light quark mass. However lit-tle has been said about this experimentally, for k ∼ 1.5GeV, where we already have m(k)≪ k.In Hamiltonian dynamics, the quark mass appears inthe Dirac spinors Uκλ that govern the quark-quark inter-action. In the heavy quark limit, spin-tensor potentialshave been successfully derived by expanding in orders ofk/m(k). For light quarks the opposite, ultrarelativisticlimit of k ≫ m(k) is relevant. Spinors are then conve-niently expanded in the inverse ratio m(k)/k, orUκλ =1√2E(k)[ √E(k) +m(k)χλ√E(k)−m(k)~σ · κˆχλ](1)−→k→∞1√2[χλ~σ · κˆχλ]+12√2m(k)k[χλ−~σ · κˆχλ]with E(k) =√k2 +m(k)2, having kept the leading chi-rally invariant term, and a next order chiral symmetrybreaking m(k)kterm. Non-chiral, spin-dependent poten-tials in the quark-quark interaction originate from the2second term in an expansion of HQCD [13] in the weaksense (that is, not of the Hamiltonian operator itself, buta restriction thereof to the Hilbert space of highly excitedresonances, where 〈k〉 is large):〈n|HQCD|n′〉 ≃ 〈n|HQCDχ |n′〉+〈n|m(k)kHQCD′χ |n′〉+ . . .(2)To access the quark mass from experiment one can ex-ploit the approximate parity degeneracy, that would beexact in the presence of only the first term HQCDχ , of ex-cited baryons. This degeneracy follows from invarianceunder transformations generated by the chiral charge [14]Qa5 =∫dxψ†(x)γ5τa2 ψ(x) and [Qa5, H ] = 0. HoweverChiral Symmetry is spontaneously broken by the groundstate, Qa5|0〉 6= 0 providing a large quark mass in thepropagator, m(k), pseudo-Goldstone bosons (π, K, η),and the loss of parity degeneracy in ground state baryons.Substituting the spinors, and in terms of Bogoliubov-rotated qq¯ normal modes B and D, Q5 becomesQa5 =∫d3k(2π)3∑λλ′ff ′c(τa2)ff ′k√k2 +m2(k)(3)((σ ·kˆ)λλ′(B†kλfcBkλ′f ′c +D†−kλ′f ′cD−kλfc)+m(k)k(iσ2)λλ′(B†kλfcD†−kλ′f ′c +Bkλ′f ′cD−kλfc)).In the presence of Spontaneous χSB, m(k) 6= 0, andthe last term realizes chiral symmetry non-linearly inthe spectrum as it creates/destroys a pion. But when〈k〉 ≫ m(k), the σ·kˆ-term dominates and chiral symme-try is realized linearly (with only quark counting opera-tors flipping parity and spin). This happens for a highlyexcited baryon resonance whose constituents have a mo-mentum distribution peaked at higher momenta than theIR enhancement of m(k) (Fig. 1).For the ground–state resonances in excited spin chan-nels, three–quark variational wavefunctions are relevantand a reasonable phenomenological guide. When Q5 actson such wavefunction |N〉 = FijkB†iB†jB†k|Ω〉, the re-sult also contains three quarks, but one of them is spin-rotated from Bkλ to σ · kˆλλ′Bkλ′ . Successive applicationof the chiral charge spin-rotates further quarks, changingeach time the parity of the total wavefunction. Howeverthe sequence of states is closed since σ·kˆσ·kˆ = I. In fact,starting with an arbitrary spin–symmetric wavefunctionwith parity P , one generates our claimed Baryon Quartetof equal isospin (isospin index dropped):|NP0 〉 =∑FPijkB†iB†jB†k|Ω〉 (4)|N−P1 〉 =13∑FPijk((σ ·kˆB†)iB†jB†k+B†i(σ ·kˆB†)jB†k +B†iB†j(σ ·kˆB†)k)|Ω〉|NP2 〉 =13∑FPijk((σ ·kˆB†)i(σ ·kˆB†)jB†k+B†i(σ ·kˆB†)j(σ ·kˆB†)k+(σ ·kˆB†)iB†j(σ ·kˆB†)k)|Ω〉|N−P3 〉 =∑FPijk(σ ·kˆB†)i(σ ·kˆB†)j(σ ·kˆB†)k|Ω〉which is a natural basis to discuss chiral symmetryrestoration in baryons, through wavefunctions that arelinear combinations |N〉 =∑ ca|Na〉. This reducible rep-resentation of Qa=35 corresponds to [(3/2, 0) ⊕ (0, 3/2)]2of SU(2)L×SU(2)R, and can be embedded in larger onesincluding nucleons [7] .Because [Q5, H ] = 0, the two can be diagonalized si-multaneously. The representation of the chiral charge inthe (non-orthonormal) quartet coordinates breaks in twoblocks since Q5 changes parity, or using the square chargeQ25(c0c2)=[3 26 7](c0c2). (5)Immediately one sees that the two linear combinationsN0−N2 andN0+3N2 diagonalize the square chiral chargein the positive parity sector (if the original wavefunctionhad positive parity), with N1 −N3 and 3N1 +N3 doingso for negative parity. The quartet then separates intotwo doublets connected by the chiral chargeQ5(N0 −N2) = N1 −N3 Q5(N0 + 3N2) = 3(3N1 +N3)Q5(N1 −N3) = N0 −N2 Q5(3N1 +N3) = 3(N0 + 3N2) .(6)Two doublets appear with different eigenvalues of Q25, 1and 9 respectively, and the interdoublet splitting becomesa dynamical question (we will argue shortly that it issmall for highly excited baryons). However, the splittingwithin the doublet must vanish asymptotically. Even forfixed (not running) quark mass, when the typical kineticenergy is high enough 〈k〉 >> m, the effects of the quarkmass are negligible. Parity doubling then boils down towhether the interaction terms are also chiral symmetryviolating or not.We now exploit the smallness of the |MP=+−MP=−|mass difference with increasing angular momentum toguide experiment in obtaining information about therunning quark mass. For this we establish how the j-dependence of the splitting is related to the k-dependenceof m(k). The M± in our proposed study are the massesof the ground state quartets of the ∆ spectrum, with par-ity ± and in the limit of large total angular momentumj >> 3/2. The two approximately degenerate massesM+ and M− are in the same leading linear Regge tra-jectory, phenomenologically fixing their j-dependence toj = α0 + αM±2 −→j→∞αM±2. (7)The parity of the ground state alternates between + and− as the angular momentum steps up by one. Large j32 4 6k (GeV)00.10.20.30.40.5Lattice m(k) (GeV) m (k)∆3/2∆5/2∆7/2∆9/2∆11/2∆13/2FIG. 2: Typical momentum distributions of increasinglyexcited ∆3/2, . . .∆13/2 resonances overlap less and lesswith the dynamically generated IR quark mass so that〈n|m(k)kHQCD′χ |n〉 becomes small. (Illustrative variationalwavefunctions for a linear potential with string tension σ =0.135 GeV2, not normalized for visibility).is equivalent to large quark orbital angular momentuml ≫ 1 since the spin is finite, and also to a large averagelinear momentum 〈k〉 ≫ ΛQCD. Figure 2 shows how thewavefunctions of high-j states overlap with the runningm(k). In these states 〈m(k)/k〉 is a useful perturbativeparameter. From the relativistic virial theorem [15] be-tween kinetic and total energies,〈k〉 → c2M± → c2√α√j , (8)(where c2 is a constant; for instance in the case of a linearpotential model, and 3 quarks in a baryon, c2 = 1/6).The first term in Eq. (2) cancels out in the difference|M+ −M−| << M± (while both M± are dominated bythe chirally invariant term, their difference stems fromthe dynamically generated quark mass) thus exposing thesecond term in eq.(2), proportional to 〈m(k)k〉, viz.|M+−M−|→〈m(k)kHQCD′χ 〉 → c3m(〈k〉)〈k〉 〈HQCD ′χ 〉 (9)(the factorization is allowed by the mean-value theoremat the price of a constant that we do not attempt to deter-mine here). This equation is analogous to the renownedGell-Mann-Oakes-Renner relation M2pi = −mq 〈ψ¯ψ〉f2pi butactive when chiral symmetry is realized linearly, as inthe high-baryon excitations we examine.Notice that HQCD′χ differs from HQCDχ due to the −sign of the (σ· kˆ)i term in the 〈m(k)/k〉 expansion of thequark i spinor in eq. (2). This changes the sign of spin-dependent terms in HQCD′χ , led in high j by the spin-orbit term. In HQCDχ the spin-orbit is crucial to correctthe angular momentum in the centrifugal barrier termfrom Li2 to the chirally invariant Li2+2Li ·Si = Ji2− 34 .But the opposite sign of the spin-orbit term of HQCD′χproduces a |M+−M−| mass difference, enhanced in theIR by 〈m(k)/k〉. Since the centrifugal barrier scales in1 2 3 4 5 6 7 8 9J11.522.533.54M (GeV)Computed, natural parityComputed, parity doublerPDG, natural parityPDG, parity doublerFIG. 3: Parity doubling in the spin-excited ∆ spectrum fora string tension σ = 0.135 GeV2 and the mass gap anglematched to the lattice data. A qqq variational Monte-Carlocomputation of the masses with a chiral Hamiltonian showsthat the ground-states with parity ± for each j = 3/2 . . . 13/2quickly degenerate. Experimentally the degeneracy can beclaimed for the 9/2 states only.the large j limit like the baryon mass M±, and the spin-orbit term comes with one power of j less than it,〈HQCD ′χ 〉 → c5M±j−1 →c5√α√1j. (10)Combining the j-power-law analysis of eqs. (7) to (10),|M+ −M−|→ c3c5c2√αm(〈k〉) j−1 (11)that links the IR enhancement of the quark mass tobaryon spectroscopy in a usable way. An experimentalextraction proceeds by just fitting the exponent −i of jin the splitting |M+ −M−| ∝ j−i. Then, we obtain,m(c2√α×√j) ∝ j−i+1 and m(k) ∝ k−2i+2 . (12)The same exponent i in eq. (12) can be obtained fromthe fit [16] to the |M+−M−| with increasing j! Thoughthe quark mass itself is gauge–dependent, our analysissuggests that its power–law exponent is not, since it isdirectly related to an observable.We next contribute the first competitive chirally in-variant quark model computation of parity doubling inthe excited baryon spectrum, shown in figure 3, employ-ing a well-known model of Coulomb-QCD that can beseen as a field theory upgrade of the Cornell model,H = −gs∫dxΨ†(x)α·A(x)Ψ(x) + Tr∫dx(E·E+B·B)+∫dxΨ†q(−iα·∇+ βm)Ψq−12∫dxdyρaxVL(|x− y|)ρaywith the kernel VL(r) = σr with string tension σ = 0.135GeV2, coupled to the color charge density ρa(x) =Ψ†(x)T aΨ(x) + fabcAb(x) · Πc(x) . We solve the BCSgap equation and employ both the so calculated m(k)40 1 2 3 4 5 6 7J00.10.20.30.40.50.60.70.80.91|E + - E |- (GeV)Doublet splitting0 1 2 3 4 5 6 7J00.10.20.30.40.50.60.70.80.91|E p - E’| p (GeV)Interdoublet: natural parityinterdoublet: unnatural parityFIG. 4: All model quartet splittings decrease for high j. Left:intradoublet splitting for the first parity doublet. Right: in-terdoublet splittings between natural and unnatural paritystates. The latter is smaller for low j.quark mass as well as lattice computation in Landauand Coulomb gauges [11]. The model has the samechiral structure of QCD[4], satisfying the Gell-Mann-Oakes-Renner relation and other low-energy theorems,and allowing computations of static Nπ observables [17].Pauli antisymmetrization of wavefunctions, spin summa-tions and the 9-dimensional integrations are performedin a computer program, employing Monte-Carlo integra-tion [18], to compute the excited ∆∗ spectrum. We pro-ceed variationally and employ several types of wavefunc-tions. While a suboptimal variational basis for low-lyingstates, the quartet states coincide with H-eigenstateshigh in the spectrum. We use both the quartet and thefixed L variational basis.As can be seen from figure 4, within our variationaland Monte-Carlo integration errors, the model splittingsdrop with j, as predicted analytically; a model statementequivalent to the generic eq. (9) readsM+ −M− = 3∫d3k1(2π)3d3k2(2π)3(23)∫d3q(2π)3(13)Vˆ (q)12(m(|k1|)|k1| +m(|k1 + q|)|k1 + q|)F ∗λ1λ2λ3(k1,k2)(I− σkˆ1σk̂1 + q)λ1µ1Fµ1λ2λ3(k1 + q,k2 − q)If precise data becomes available at ELSA or JeffersonLab for the ∆J with J = 7/2, 9/2, 11/2, 13/2 · · · par-ity quartets, one should be able to distinguish betweenthe 1/j fall of |M+ −M−| for non-chiral models (witha constant difference (M+ − M−)2 between the ReggeTrajectories [10]), and the faster drops for chiral theoriessuch as QCD. Since the two doublets are closely degener-ate, both positive and negative parity ground states willhave a nearby resonance with identical quantum num-bers. Given their width, it is likely they will only bedistinguished by very careful exclusive decay analysis.Further, if the ∆ spectrum could be measured up tohigh resonance masses, beyond the IR m(k) enhance-ment, and a lattice calculation of 〈HQCD ′χ 〉 became avail-able, an almost direct measurement of the current quarkmass would follow in a regime of logarithmic, rather thanpower-law running. We advocate that a measurementof masses of high partial-wave ∆ resonances with anaccuracy of 50 MeV should suffice to establish the ap-proximate degeneracy and test the concept of runningquark mass in the IR. The presented concepts shouldalso further motivate analysis of empirical data[19] suchas EBAC (Excited Baryon Analysis Center) at JeffersonLab as well as Lattice QCD collaborations who are ad-dressing the baryon spectrum.We thank L. Glozman for useful conversations,and grants FPA 2008-00592/FPA, FIS2008-01323,CERN/FP /83582/2008, POCI/FP /81933/2007,/81913/2007 , PDCT/FP /63907/2005 and /63923/2005,Spain-Portugal billateral grant HP2006-0018 / E-56/07,as well as the Scientific Research Fund of Flanders.[1] L. Y. Glozman, Phys. Lett. B 475 (2000) 329.[2] E. S. Swanson, Phys. Lett. B 582, 167 (2004).[3] R. F. Wagenbrunn and L. Y. Glozman, Phys. Lett.B 643, 98 (2006); T. D. Cohen and L. Y. Glozman,Mod. Phys. Lett. A 21, 1939 (2006); L. Y. Glozman,A. V. Nefediev and J. E. F. Ribeiro, Phys. Rev. D 72,094002 (2005).[4] A. Le Yaouanc et al., Phys. Rev. D 31 (1985) 137.[5] P. Bicudo et al., Phys. Lett. B 442, 349 (1998).[6] C. E. DeTar and T. Kunihiro, Phys. Rev. D 39, 2805(1989); D. Jido, T. Hatsuda and T. Kunihiro, Phys. Rev.Lett. 84, 3252 (2000).[7] T. D. Cohen and L. Y. Glozman, Phys. Rev. D 65,016006 (2001) [arXiv:hep-ph/0102206].[8] S. Sarkar et al., arXiv:0902.3150 [hep-ph].[9] C. Amsler et al. [PDG], Phys. Lett. B 667, 1 (2008).[10] J. Segovia, D. R. Entem and F. Fernandez, Phys. Lett.B 662, 33 (2008).[11] P. O. Bowman, et al., Nucl. Phys. Proc. Suppl. 161, 27(2006); M. B. Parappilly et al, Phys. Rev. D 73, 054504(2006). S. Furui, arXiv:0801.0325 [hep-lat].[12] R. Alkofer et al., Ann. Phys. NY (in press)arXiv:0804.3042 [hep-ph].[13] For a discussion on HQCD, a good starting point is N. H.Christ and T. D. Lee, Phys. Rev. D22 (1980) 939.[14] A. V. Nefediev, J. E. F. Ribeiro and A. P. Szczepaniak,JETP Lett. 87, 271 (2008).[15] W. Lucha and F. F. Schoberl, Mod. Phys. Lett. A 5, 2473(1990).[16] For a linear potential, with m(k) ∝ k−4, |M+ −M−| ∝1j3. If, on the contrary, the quark mass is constant, andstill the potential remains chirally invariant, then the de-crease follows a slower j−1. We can retrospectively un-derstand analytically the qq¯ numerical results of [3], withchiral quartets and with the j-scaling of the splittings.While their doublet |M+−M−| ∼ 1j3; their interdoubletsplitting follows a weaker 1j3/2as expected in a corollaryof eq. (10) for spin-independent potentials.[17] P. Bicudo, G. Krein and J. E. F. Ribeiro, Phys. Rev. C64 (2001) 025202.[18] T. Hahn, Comput. Phys. Commun. 168, 78 (2005).[19] B. Julia-Diaz et al, Phys. Rev. C 77 (2008) 045205.

© 2009 The American Physical Society.
We thank L. Glozman for useful conversations and grants FPA 2008-00592/FPA, FIS2008-01323, CERN/FP/83582/2008, POCI/FP/81933/2007, /81913/2007, PDCT/FP/63907/2005, and /63923/2005, Spain-Portugal billateral grant HP2006-0018/E-56/07, as well as the Scientific Research Fund of Flanders.We argue that three-quark excited states naturally group into quartets, split into two parity doublets, and that the mass splittings between these parity partners decrease higher up in the baryon spectrum. This decreasing mass difference can be used to probe the running quark mass in the midinfrared power-law regime. A measurement of masses of high-partial-wave  resonances should be sufficient to unambiguously establish the approximate degeneracy. We test this concept with the first computation of excited high-j baryon masses in a chirally invariant quark model.Spain-Portugal billateralScientific Research Fund of FlandersFPADepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Llanes Estrada, Felipe José

Cardoso, M

Docta Complutense

Probing the Infrared Quark Mass from Highly Excited BaryonsP. Bicudo,1 M. Cardoso,1 T. Van Cauteren,2 and Felipe J. Llanes-Estrada31CFTP, Departamento de Fı́sica, Instituto Superior Técnico, Lisboa, Portugal2Department of Subatomic and Radiation Physics, Ghent University, Ghent, Belgium3Departamento de Fı́sica Teórica I, Universidad Complutense de Madrid, Madrid, Spain(Received 8 March 2009; published 25 August 2009)We argue that three-quark excited states naturally group into quartets, split into two parity doublets, andthat the mass splittings between these parity partners decrease higher up in the baryon spectrum. Thisdecreasing mass difference can be used to probe the running quark mass in the midinfrared power-lawregime. A measurement of masses of high-partial-wave  resonances should be sufficient to unambig-uously establish the approximate degeneracy. We test this concept with the first computation of excitedhigh-j baryon masses in a chirally invariant quark model.DOI: 10.1103/PhysRevLett.103.092003 PACS numbers: 12.39.Ki, 11.30.Fs, 12.38.AwQuantum chromodynamics (QCD) has been thoroughlytested in high-energy physics, through hadron jets, Drell-Yan processes, eeþ annihilation, and deep inelastic scat-tering. Low-energy QCDmanifests chiral symmetry break-ing (SB) that enhances quark masses in the infrared (IR),generating most of the visible mass in our Universe, andthat removes the degeneracy from the ground states of lighthadron spectra (so light hadrons do not come in paritydoublets). The excited light-quark baryon spectrum, withmasses above that of the  resonance (1232 MeV), has notbeen accessible to any form of perturbation theory, effec-tive or fundamental.Insensitivity to chiral symmetry breaking, recentlystressed by Glozman and co-workers [1,2] and Swanson[3], and retrospectively present in chirally invariant quarkmodels [4,5] has led many hadron physicists to accept thatspontaneous SB, the salient feature of the low hadronspectrum, becomes less important for highly excited reso-nances. For these, chiral symmetry is asymptotically real-ized in Wigner mode, arranging hadrons in degeneratechiral multiplets [6]. We have realized that a formalizationof this statement is that the ratio of the quark mass to itsmomentum hmk i provides a new perturbative parameter tostudy some aspects of the excited hadron spectrum whenthe typical momentum becomes higher than the runningmass. Reasoning in reverse, we propose employing a can-cellation in mass differences of very excited resonances, toaccess this small parameter, and hence the quark mass. Wealso show analytically and numerically that three-quarkstates naturally group into quartets with two states ofeach parity. Diagonalizing the chiral charge in terms ofquarks, the quartet is split into two parity doublets, and allmass splittings tend to decrease when going higher in thespectrum. For simplicity and to minimize the impact ofmolecular meson-nucleon configurations [7], we study themaximum-spin excitations  of the Delta baryon.The u or d quark mass [8] (understood as taken from thefull two-point function of the field theory) is theorized tovary from m ’ 1–6 MeV at a high, perturbative momen-tum k scale (typically QCD  210 MeV) to a constitu-ent mass of circa 300 MeV at vanishing momentumk ’ QCD. This IR enhancement of 2 orders of magnitudeis shown in Fig. 1. The simplest constituent quark modelconsiders a constant mass [9], without the characteristicstep function of the Nambu and Jona-Lasinio model (alsoshown). Breaking chiral symmetry with a linear potentialforces a power law m ! Ck4for mðkÞ  k. Lattice QCD[10] and Schwinger-Dyson equations [11], both Euclidianapproaches to QCD, also show power-law decays of thelight-quark mass. However, little has been said about thisexperimentally, for k 1:5 GeV, where we already havemðkÞ  k.In Hamiltonian dynamics, the quark mass appears in theDirac spinors U that govern the quark-quark interaction.In the heavy quark limit, spin-tensor potentials have beensuccessfully derived by expanding in orders of k=mðkÞ. Forlight quarks the opposite, ultrarelativistic limit of k mðkÞ is relevant. Spinors are then conveniently expanded0 2000 4000k (MeV)0100200300400m(k) (MeV)Constituent  Quark ModelNambu-Jona-LasinioLinear confinement Landau gauge latticeCoulomb gauge latticeDyson-SchwingerFIG. 1 (color online). IR enhancement of the light-quark massdue to spontaneous SB. Shown are quark masses in the mainapproaches to QCD, all multiplied by an arbitrary factor tomatch them at quark momentum norm k ! 0. (References inthe text.)PRL 103, 092003 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending28 AUGUST 20090031-9007=09=103(9)=092003(4) 092003-1  2009 The American Physical Societyin the inverse ratio mðkÞ=k, orU ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi2EðkÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðkÞ þmðkÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðkÞ mðkÞp ~  ̂24 35!k!11ffiffiffi2p ~  ̂" #þ 12ffiffiffi2p mðkÞk ~  ̂" #;(1)with EðkÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þmðkÞ2p , having kept the leading chirallyinvariant term, and a next order chiral symmetry breakingmðkÞk term. Nonchiral, spin-dependent potentials in thequark-quark interaction originate from the second term inan expansion ofHQCD [12] in the weak sense (that is, not ofthe Hamiltonian operator itself, but a restriction thereof tothe Hilbert space of highly excited resonances, where hki islarge):hnjHQCDjn0i ’ hnjHQCD jn0i þ hnjmðkÞkHQCD0 jn0i þ    :(2)To access the quark mass from experiment one can exploitthe approximate parity degeneracy, that would be exact inthe presence of only the first term HQCD , of excited bary-ons. This degeneracy follows from invariance under trans-formations generated by the chiral charge [13]Qa5 ¼Rdxc yðxÞ5 a2 c ðxÞ and ½Qa5 ; H ¼ 0. However,chiral symmetry is spontaneously broken by the groundstate, Qa5j0i  0, providing a large quark mass in thepropagator, mðkÞ, pseudo-Goldstone bosons (;K; ),and the loss of parity degeneracy in ground state baryons.Substituting the spinors, and in terms of Bogoliubov-rotated q q normal modes B and D, Qa5 becomesQa5 ¼Z d3kð2Þ3X0ff0ca2ff0kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þm2ðkÞpð  k̂Þ0 ðBykfcBk0f0cþDyk0f0cDkfcÞþmðkÞkði2Þ0 ðBykfcDyk0f0cþBk0f0cDkfcÞ: (3)In the presence of spontaneous SB,mðkÞ  0, and the lastterm realizes chiral symmetry nonlinearly in the spectrumas it creates or destroys a pion. But when hki  mðkÞ, the  k̂ term dominates and chiral symmetry is realizedlinearly (with only quark counting operators flipping parityand spin). This happens for a highly excited baryon reso-nance whose constituents have a momentum distributionpeaked at higher momenta than the IR enhancement ofmðkÞ (Fig. 2).For the ground-state resonances in excited spin chan-nels, three-quark variational wave functions are relevantand a reasonable phenomenological guide. When Qa5 actson such wave function jNi ¼ FijkByi Byj Byk ji, the resultalso contains three quarks, but one of them is spin rotatedfrom Bk to   k̂0Bk0 . Successive application of thechiral charge spin rotates further quarks, changing eachtime the parity of the total wave function. However, thesequence of states is closed since   k̂  k̂ ¼ I. Startingwith an arbitrary antisymmetric wave function FPijk withparity P, one generates our claimed baryon quartet of equalisospin (isospin index dropped):jNP0 i ¼XFPByi Byj Byk jijNP1 i ¼XFPijkð  k̂ByÞiByj BykijNP2 i ¼XFPijkð  k̂ByÞið  k̂ByÞjBykijNP3 i ¼XFPijkð  k̂ByÞið  k̂ByÞjð  k̂ByÞkji;(4)which is a natural basis to discuss chiral symmetry resto-ration in baryons. This reducible representation of Qa5corresponds to ½ð3=2; 0Þ 	 ð0; 3=2Þ2 of SUð2ÞL SUð2ÞR, and can be embedded in larger ones includingnucleons [6]. Because ½Qa5 ; H ¼ 0, the two can be diago-nalized simultaneously. The representation of the chiralcharge in the quartet in Eq. (4) breaks in two blocks sinceQa5 changes parity, or using the square chargeQa25N0N2¼3 26 7N0N2: (5)Immediately one sees that the two linear combinationsN0  N2 and N0 þ 3N2 diagonalize the square chiralcharge in the positive parity sector (if the original wavefunction had positive parity), with N1  N3 and 3N1 þ N3doing so for negative parity. The quartet then separates intotwo doublets connected by the chiral charge Qa5ðN0 N2Þ ¼ N1  N3 and Qa5ðN0 þ 3N2Þ ¼ 3ðN1 þ N3Þ withdifferent eigenvalues of Qa25 , 1 and 9, respectively. The2 4 6k (GeV)00.10.20.30.40.5Lattice m(k) (GeV) m (k)∆3/2∆5/2∆7/2∆9/2∆11/2∆13/2FIG. 2 (color online). Typical momentum distributions of in-creasingly excited 3=2; . . . ;13=2 resonances overlap less andless with the dynamically generated IR quark mass so thathnj mðkÞk HQCD0 jni becomes small. (Illustrative variational wavefunctions for a linear potential with string tension  ¼0:135 GeV2, not normalized for visibility.)PRL 103, 092003 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending28 AUGUST 2009092003-2interdoublet splitting becomes a dynamical question (wewill argue shortly that it is small for highly excited bary-ons). However, the splitting within the doublet must vanishwhen the kinetic energy is high enough hki  m. Paritydoubling then boils down to whether the interaction termsare also chiral symmetry violating or not.We now exploit the smallness of the jMP¼þ MP¼jmass difference with increasing angular momentum toguide experiment in obtaining information about the run-ning quark mass. For this we establish how the j depen-dence of the splitting is related to the k dependence ofmðkÞ. The M in our proposed study are the masses of theground state quartets of the  spectrum, with parity andin the limit of large total angular momentum j  3=2. Thetwo approximately degenerate masses Mþ and M are inthe same leading linear Regge trajectory, phenomenologi-cally fixing their j dependence toj ¼ 	0 þ 	M2 !j!1 	M2: (6)The parity of the ground state alternates between þ andas the angular momentum steps up by one. Large j isequivalent to large quark orbital angular momentum l 1 since the spin is finite, and also to a large average linearmomentum hki  QCD. Figure 2 shows how the wavefunctions of high-j states overlap with the runningmðkÞ. Inthese states hmðkÞ=ki is a useful perturbative parameter.From the relativistic virial theorem [14] between kineticand total energies,hki ! c2M ! c2ffiffiffiffi	p ffiffijp (7)(where c2 is a constant; for instance in the case of a linearpotential model, and 3 quarks in a baryon, c2 ¼ 1=6). Thefirst term in Eq. (2) cancels out in the difference jMþ Mj  M (while bothM are dominated by the chirallyinvariant term, their difference stems from the dynamicallygenerated quark mass), thus exposing the second term inEq. (2), proportional to hmðkÞk i, viz.jMþ Mj !mðkÞkHQCD0! c3 mðhkiÞhki hHQCD0 i (8)(the factorization is allowed by the mean-value theorem atthe price of a constant that we do not attempt to determinehere). This equation is analogous to the renowned Gell-Mann-Oakes-Renner relation M2 ¼ mq h c c if2 but activewhen chiral symmetry is realized linearly, as in the high-baryon excitations we examine. Notice that HQCD0 differsfrom HQCD due to the  sign of the ð  k̂Þi term in thehmðkÞ=ki expansion of the quark i spinor in Eq. (2). Thischanges the sign of spin-dependent terms in HQCD0 , led inhigh j by the spin-orbit term. In HQCD the spin orbit iscrucial to correct the angular momentum in the centrifugalbarrier term from L2i to the chirally invariant L2i þ 2Li Si ¼ J2i  34 . But the opposite sign of the spin-orbit term ofHQCD0 produces a jMþ Mj mass difference, enhancedin the IR by hmðkÞ=ki. Since the centrifugal barrier scalesin the large j limit like the baryon mass M, and the spin-orbit term comes with one power of j less than it,hHQCD0 i ! c5Mj1 ! c5ffiffiffiffi	pffiffiffi1js: (9)Combining the j-power-law analysis of Eqs. (6)–(9),jMþ Mj ! c3c5c2ffiffiffiffi	p mðhkiÞj1; (10)that links the IR enhancement of the quark mass to baryonspectroscopy in a usable way. An experimental extractionproceeds by just fitting the exponenti of j in the splittingjMþ Mj / ji. Then, we obtainmc2ffiffiffiffi	p  ffiffijp  / jiþ1 and mðkÞ / k2iþ2: (11)The same exponent i in Eq. (11) can be obtained from thefit [15] to the jMþ Mj with increasing j. Though thequark mass itself is gauge-dependent, our analysis suggeststhat its power-law exponent is not, since it is directlyrelated to an observable.We next contribute the first competitive chirally invari-ant quark model computation of parity doubling in theexcited baryon spectrum, shown in Fig. 3, employing awell-known model of Coulomb QCD that can be seen as afield theory upgrade of the Cornell model,H¼gsZdxyðxÞ AðxÞðxÞþTrZdxðE EþB BÞþZdxyq ði rþmÞq12ZdxdyaxVLðjxyjÞay ;1 2 3 4 5 6 7 8 9J11.522.533.54M (GeV)Computed, natural parityComputed, parity doublerPDG, natural parityPDG, parity doublerFIG. 3 (color online). Parity doubling in the spin-excited spectrum for a string tension  ¼ 0:135 GeV2 and the mass gapangle matched to the lattice data. A qqq variational Monte Carlocomputation of the masses with a chiral Hamiltonian shows thatthe ground states with parity  for each j ¼ 3=2; . . . ; 13=2quickly degenerate. Experimentally the degeneracy can beclaimed for the 9=2 states only, as reported by the ParticleData Group (PDG) [8].PRL 103, 092003 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending28 AUGUST 2009092003-3with the kernel VLðrÞ ¼ r with string tension  ¼0:135 GeV2, coupled to the color charge density aðxÞ ¼yðxÞTaðxÞ þ fabcAbðxÞ cðxÞ. We solve the BCSgap equation and employ both the so-calculatedmðkÞ quarkmass as well as lattice computation in Landau andCoulomb gauges [10]. The model has the same chiralstructure of QCD [4], satisfying the Gell-Mann-Oakes-Renner relation and other low-energy theorems, and allow-ing computations of static N observables [16]. Pauliantisymmetrization of wave functions, spin summations,and the nine-dimensional integrations are performed in acomputer program, employing Monte Carlo integration[17], to compute the excited  spectrum. We proceedvariationally with several types of wave functions. Whilea suboptimal variational basis for low-lying states, thequartet states coincide with H eigenstates high in thespectrum. As can be seen from Fig. 4, within our varia-tional and Monte Carlo integration errors, the model split-tings drop with j, as predicted analytically. A modelstatement equivalent to the generic Eq. (8) readsMþM¼3Z d3k1ð2Þ3d3k2ð2Þ323Z d3qð2Þ3 V̂ðqÞ12mðjk1jÞjk1j þmðjk1þqjÞjk1þqjF123ðk1;k2ÞðI  k̂1  dk1þqÞ11F123ðk1þq;k2qÞ: (12)If precise data becomes available at ELSA or JeffersonLab for the J with J ¼ 7=2; 9=2; 11=2; 13=2; . . . parityquartets, one should be able to distinguish between the 1=jfall of jMþ Mj for nonchiral models (with a constantdifference ðMþ MÞ2 between the Regge trajectories[9]), and the faster drops for chiral theories such as QCD.Since the two doublets are closely degenerate, both posi-tive and negative parity ground states will have a nearbyresonance with identical quantum numbers. Given theirwidth, it is likely they will only be distinguished by verycareful exclusive decay analysis. Further, if the spectrumcould be measured up to high resonance masses, beyondthe IR mðkÞ enhancement, and a lattice calculation ofhHQCD0 i became available, an almost direct measurementof the current quark mass would follow in a regime oflogarithmic, rather than power-law running. We advocatethat a measurement of masses of high partial-wave resonances with an accuracy of 50 MeV should suffice toestablish the approximate degeneracy and test the conceptof running quark mass in the IR. The presented conceptsshould also further motivate analysis of empirical data [18]such as Excited Baryon Analysis Center (EBAC) atJefferson Lab as well as lattice QCD collaborations.We thank L. Glozman for useful conversations andgrants FPA 2008-00592/FPA, FIS2008-01323, CERN/FP/83582/2008, POCI/FP/81933/2007, /81913/2007, PDCT/FP /63907/2005, and /63923/2005, Spain-Portugal billa-teral grant HP2006-0018/E-56/07, as well as the ScientificResearch Fund of Flanders.[1] L. Y. Glozman, Phys. Lett. B 475, 329 (2000).[2] R. F. Wagenbrunn and L.Y. Glozman, Phys. Lett. B 643,98 (2006); T. D. Cohen and L.Y. Glozman, Mod. Phys.Lett. A 21, 1939 (2006); L. Y. Glozman, A. V. Nefediev,and J. E. Ribeiro, Phys. Rev. D 72, 094002 (2005).[3] E. S. Swanson, Phys. Lett. B 582, 167 (2004).[4] A. Le Yaouanc et al., Phys. Rev. D 31, 137 (1985).[5] P. Bicudo et al., Phys. Lett. B 442, 349 (1998).[6] C. E. DeTar and T. Kunihiro, Phys. Rev. D 39, 2805(1989); D. Jido, T. Hatsuda, and T. Kunihiro, Phys. Rev.Lett. 84, 3252 (2000).[7] S. Sarkar et al., arXiv:0902.3150.[8] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1(2008).[9] J. Segovia, D. R. Entem, and F. Fernandez, Phys. Lett. B662, 33 (2008).[10] P. O. Bowman et al., Nucl. Phys. B, Proc. Suppl. 161, 27(2006); M.B. Parappilly et al., Phys. Rev. D 73, 054504(2006); S. Furui, Few-Body Syst. 45, 51 (2009); 46, 73(2009).[11] R. Alkofer et al., Ann. Phys. (N.Y.) 324, 106 (2009).[12] For a discussion on HQCD, a good starting point is N. H.Christ and T. D. Lee, Phys. Rev. D 22, 939 (1980).[13] A. V. Nefediev, J. E. Ribeiro, and A. P. Szczepaniak, JETPLett. 87, 271 (2008).[14] W. Lucha and F. F. Schoberl, Mod. Phys. Lett. A 5, 2473(1990).[15] For a linear potential, with mðkÞ / k4, jMþ Mj / 1j3.If, on the contrary, the quark mass is constant, and still thepotential remains chirally invariant, then the decreasefollows a slower j1. We can retrospectively understandanalytically the q q numerical results of [2], with chiralquartets and with the j scaling of the splittings. While theirdoublet jMþ Mj  1j3, their interdoublet splitting fol-lows a weaker 1j3=2as expected in a corollary of Eq. (9) forspin-independent potentials.[16] P. Bicudo, G. Krein, and J. E. Ribeiro, Phys. Rev. C 64,025202 (2001).[17] T. Hahn, Comput. Phys. Commun. 168, 78 (2005).[18] B. Julia-Diaz et al., Phys. Rev. C 77, 045205 (2008).0 1 2 3 4 5 6 7J00.10.20.30.40.50.60.70.80.91|E+ - E | - (GeV)Doublet splitting0 1 2 3 4 5 6 7J00.10.20.30.40.50.60.70.80.91|Ep - E’| p (GeV)Interdoublet: natural parityinterdoublet: unnatural parityFIG. 4 (color online). All model quartet splittings decrease forhigh j. Left: Intradoublet splitting for the first parity doublet.Right: Interdoublet splittings between natural and unnaturalparity states. The latter is smaller for low j.PRL 103, 092003 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending28 AUGUST 2009092003-4

Physical Review Letters

Probing the infrared quark mass from highly excited baryons

https://biblio.ugent.be/publication/749350/file/749390.pdf

Probing the infrared quark mass from highly excited baryons

Abstract

Similar works

Full text

Available Versions

Crossref

Ghent University Academic Bibliography

Docta Complutense