We observe that excited hadrons provide an opportunity to probe from
experiment the power-law running of the quark mass in the mid infrared, while
the condition m(k)<k remains valid. A relatively clean analysis is possible for
the maximum spin excitations of the Delta baryons, analogous to the yrast
states in nuclei. Such states are accessible at current experimental facilities
such as ELSA and Jlab.Comment: 6 pages, 2 figures, proceedings of ExcitedQCD, Zakopane, Poland

Bicudo, Pedro

Cardoso, Marco

Llanes-Estrada, Felipe J.

Van Cauteren, Tim

Acta Physica Polonica B Proceedings Supplement

English

arXiv

We would like to remark the very pleasant and intellectually stimulat- ing atmosphere of the meeting in Zakopane. 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 Fund for Scientific Research Flanders.
International Meeting on Excited QCD (2009. Zakopane,Polonia)We observe that excited hadrons provide an opportunity to probe from experiment the power-law running of the quark mass in the mid infrared, while the condition m(k) < k remains valid. A relatively clean analysis is possible for the maximum spin excitations of the Delta baryons accessible at current experimental facilities such as ELSA and Jlab.Fund for Scientific Research FlandersDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Llanes Estrada, Felipe José

Docta Complutense

arXiv:0905.4124v1  [nucl-th]  26 May 2009Excited baryons as experimental probes of the quark massFelipe J. Llanes-EstradaUniversidad Complutense de Madrid, SpainandPedro Bicudo, Marco CardosoInstituto Superior Técnico, Lisboa, PortugalandTim van CauterenDept. Sub. and Rad. Phys., Ghent University, BelgiumWe observe that excited hadrons provide an opportunity to probe fromexperiment the power-law running of the quark mass in the mid infrared,while the condition m(k) < k remains valid.A relatively clean analysis is possible for the maximum spin excitations ofthe ∆ baryons, analogous to the yrast states in nuclei. Such states areaccessible at current experimental facilities such as ELSA and Jlab.PACS numbers: 11.10.St, 11.30.Rd, 14.20.-c, 14.65.BtQuark masses are fundamental parameters of the Standard Model thatappear in the Lagrangian density of Quantum Chromodynamics, whose non-interacting part readsL0QCD =∑imiψ̄ψ + i∑iψ̄ 6 ∂ψ. (1)These masses can also be read off an expansion of the quark propagator forlarge momenta in perturbation theoryS(k) =i6 k −m (2)and information about them can be gathered in several ways. For lightquarks, only very coarse information is available from sum rules and lattice(1)2 m˙q˙and˙yrast˙hadrons printed on March 13, 2013gauge theory, although quark mass ratios are somewhat better known fromchiral perturbation theory [1].As the momentum of the propagating quarks becomes smaller towardsthe QCD scale ΛQCD, the mass in the denominator runs like a power-lawS(p) =iZ(p2)6 k −m(k2) with m(k2) ∝ kα (3)as has been shown by Dyson-Schwinger studies [2] in Landau gauge andlattice gauge theory [3]. For even lower k the running mass function m(k)presumably stabilizes and gives sense to the concept of constituent quarks,but to date there is no precise dynamical meaning to the quark massesobtained from quark model fits to data in terms of the QCD parameters.What we have recently proposed, and elaborate in this article, is thatthe exponent of the power-law running in the mid-infrared is accessible toexperiment by studying excited hadrons. The key idea is to exploit therecently understood parity doubling in the high hadron spectrum, by whichexcited hadrons decouple from the pion and instead become degenerate witha partner of equal parity, the mass difference being of order the quark massthat wants to be extracted, |M+ − M−| ∝ m(〈k〉). The scale at whichthis mass is probed is the average momentum of the quark in the excitedhadron. The proportionality constant is difficult to calculate, so the absolutenormalization of the quark mass is not readily accessible, but by comparingincreasingly excited hadrons with larger and larger 〈k〉, one can extract itsrunning.0 2000 4000k (MeV)0100200300400m(k) (MeV)Constituent  Quark ModelNambu-Jona-LasinioLinear confinement Landau gauge latticeCoulomb gauge latticeDyson-SchwingerFig. 1. IR enhacement of the light quark mass, generated when spontaneous chiralsymmetry breaking occurs. Shown are quark masses in the main approaches toQCD, all multiplied by an arbitrary factor to match them at zero momentum. tm˙q˙and˙yrast˙hadrons printed on March 13, 2013 3In Hamiltonian dynamics, the runningm(k) defined from the propagatorappears in the Dirac spinors Uκλ. For light quarks one can expand spinorsin the ultrarelativistic (large-momentum) limit in the inverse ratio m(k)/kUκλ =1√2E(k)[√E(k) +m(k)χλ√E(k) −m(k)~σ · κ̂χλ](4)−→k→∞1√2[χλ~σ · κ̂χλ]+12√2m(k)k[χλ−~σ · κ̂χλ]with E(k) =√k2 +m(k)2, having kept the leading chiral invariant term,and a next order chiral symmetry breaking m(k)kterm. Non-chiral, spin-dependent potentials in the quark-quark interaction originate from the sec-ond term in the expansion Eq. (5).This translates into an expansion of HQCD in the weak sense (that is,not of the Hamiltonian operator itself, but its restriction to the Hilbertspace of highly excited resonances, where 〈k〉 is large) is:〈n|HQCD|n′〉 ≃ 〈n|HQCDχ |n′〉 + 〈n|m(k)kHQCD′χ |n′〉 + . . . (5)The parity degeneracy [4] for highly excited hadrons follows from in-variance under transformations generated by the chiral charge [7] Qa5 =∫dxψ†(x)γ5τa2 ψ(x) and [Qa5,H] = 0. For low–lying hadrons, Chiral Symme-try is spontaneously broken by the ground state, Qa5|0〉 6= 0 providing a largequark mass in the propagator, m(k), pseudo-Goldstone bosons (π, K, η),and the loss of parity degeneracy in ground state baryons. Substituting thespinors, and in terms of Bogoliubov-rotated qq̄ normal modes B, D,Qa5 =∫d3k(2π)3∑λλ′ff ′c(τa2)ff ′k√k2 +m2(k)(6)((σ ·k̂)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, and the last term realizeschiral symmetry non-linearly in the spectrum as it creates/destroys a pion.But when 〈k〉 is large, it is the σ·̂k-term that dominates, and chiral symmetryis realized linearly (with only quark counting operators flipping parity andspin). This happens for a baryon resonance high enough in the spectrum,whose constituents have a momentum distribution peaked higher than theIR enhancement of m(k) (see figure 2.4 m˙q˙and˙yrast˙hadrons printed on March 13, 20132 4 6k (GeV)00.10.20.30.40.5Lattice mq(k) (GeV) mq(k)∆3/2∆5/2∆7/2∆9/2∆11/2∆13/2Fig. 2. Typical momentum distributions of increasingly excited ∆3/2, . . . ∆13/2resonances overlap less and less with the dynamically generated IR quark mass sothat 〈n|m(k)k HQCD′χ |n〉 becomes small. (Illustrative variational wavefunctions for alinear potential with string tension σ = 0.135 GeV2, not normalized for visibility).This computation has been reported in [5].To exploit the smallness of the |MP=+ − MP=−| mass difference, weemploy increasing angular momentum and establish how the j-scaling ofthe splitting is related to the k-scaling of m(k).The M± in our proposed study are the masses of the ground state quartetsof the ∆ spectrum, with parity ± and in the limit of large total angularmomentum j >> 3/2. These states of maximum spin for a given energymay conveniently be called hadron yrast states, in analogy with their nuclearphysics counterparts (the dizziest states, from the swedish yr , dizzy). Theyhave the advantage of being cold , that is, all the excitation energy is spentin rotation and no radial nor, specially, multiquark excitations are expectedto be present.For each spin, the two quasi-degenerate masses M+ and M− fall in thesame (leading) Regge trajectory, and this fixes their j-scalingj = α0 + αM±2 −→j→∞αM±2. (7)The parity of the ground state (the lowest of the two states) alternatesbetween + an − as the angular momentum steps up by one. Large j isequivalent to large quark orbital angular momentum l since the spin is finite,and also to a large average linear momentum 〈k〉. This is illustrated in Fig 2where we show how the overlap of the wavefunction with the running m(k)blinds high-lying states to χSB. From the relativistic version of the virialtheorem [6] between kinetic and total energies,〈k〉 → c2M± →c2√α√j (8)m˙q˙and˙yrast˙hadrons printed on March 13, 2013 5(where c2 is a constant; for a linear potential model, and 3 quarks in abaryon, c2 = 1/6).The first term in Eq. (5) cancels out in the difference |M+ −M−| <<M± (while both M± are dominated by the chiral invariant term, theirdifference stems from the dynamically generated quark mass) thus exposingthe second term in eq.(5), 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 theorem at the price of aconstant that we do not attempt to determine here). This equation is anal-ogous to the renowned Gell-Mann-Oakes-Renner relation M2π = −mq 〈ψ̄ψ〉f2πbut active when chiral symmetry is realized linearly, as in the high-baryonexcitations we examine.The σ ·k̂ from the spinors induce spin-dependent potentials in HQCD ′χ .We need its large j-scaling, equivalent to large 〈k〉 and M±, and separatelyconsider its angular and radial dependences 〈HQCD ′χ 〉 ∝ 〈HQCD′χ 〉ang ×〈HQCD ′χ 〉rad. The angular matrix element generally includes a spin-orbitterm, that leads its j-counting〈HQCD ′χ 〉ang → j (10)while the centrifugal barrier dominates the radial part,l(l + 1)〈HQCD ′χ 〉rad → j2〈HQCD ′χ 〉rad → c4M±and thus 〈HQCD ′χ 〉rad ∝ j−2M±. From Eqs. (7,10),〈HQCD ′χ 〉 → c5M±j−1 →c5√α√1j. (11)(As a corollary, note that a spin-independent potential scales like 1j3/2).The result of the j-scaling analysis reads then|M+ −M−|→ c3m(〈k〉)〈k〉 × c5M± × j−1 = c3c5c2m(〈k〉)j−1. (12)that links the IR enhancement of the quark mass to baryon spectroscopyin a usable way. An experimental extraction proceeds by just fitting theexponent of the j-scaling for the splitting |M+ −M−| ∝ j−i. Then, in viewof eq. (8), one obtains bothm(Λ ×√j) ∝ j−i+1 and m(k) ∝ k−2i+2 . (13)6 m˙q˙and˙yrast˙hadrons printed on March 13, 2013The same exponent i in this last equation can be obtained from the fit tothe |M+ −M−| with increasing j.In conclusion, the scaling with j of the mass–splittings in the ∆ spectrumcan be related to the k–running of the quark mass as defined from thequark propagator. This remarkable result holds as long as the expansion inm(〈k〉)/〈k〉 makes sense, that is, for highly excited resonances.Other consequences of Insensitivity to Chiral Symmetry Breaking highin the spectrum, such as the decoupling of excited resonances from pions(for example, the widths Γ∆∗→Nπ) can likewise be exploited to probe therunning quark mass.We would like to remark the very pleasant and intellectually stimulat-ing atmosphere of the meeting in Zakopane. We thank L. Glozman foruseful 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 Fund for Scientific Research Flanders.REFERENCES[1] See review by A. Manohar and C. Sachrajda on page 551 ofC. Amsler et al. [Particle Data Group], Phys. Lett. B 667 (2008) 1.[2] R. Alkofer, W. Detmold, C. S. Fischer and P. Maris, Phys. Rev. D 70 (2004)014014 [arXiv:hep-ph/0309077].[3] P. O. Bowman, et al., Nucl. Phys. Proc. Suppl. 161, 27 (2006); M. B. Parap-pilly et al, Phys. Rev. D 73, 054504 (2006). S. Furui, arXiv:0801.0325 [hep-lat].[4] C. E. Detar and T. Kunihiro, Phys. Rev. D 39, 2805 (1989); D. Jido, T. Hat-suda and T. Kunihiro, Phys. Rev. Lett. 84, 3252 (2000); L. Y. Glozman,Phys. Lett. B 475 (2000) 329; E. S. Swanson, Phys. Lett. B 582, 167 (2004);R. F. Wagenbrunn and L. Y. Glozman, Phys. Lett. B 643, 98 (2006); T. D. Co-hen 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).[5] P. Bicudo, M. Cardoso, T. Van Cauteren and F. J. Llanes-Estrada,arXiv:0902.3613 [hep-ph].[6] W. Lucha and F. F. Schoberl, Mod. Phys. Lett. A 5, 2473 (1990).[7] A. V. Nefediev, J. E. F. Ribeiro and A. P. Szczepaniak, JETP Lett. 87, 271(2008).

Excited baryons as experimental probes of the quark mass

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Excited baryons as experimental probes of the quark mass

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