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Unquenching the Quark Model

Unquenching the Quark Model. Motivation Required ingredients Self-consistent baryon masses in presence of continuum (baryon-meson) states SU(6) (flavor-spin) symmetry Minimum set of intermediate states Prior and ongoing work Lessons from covariant calculation of  - 

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Unquenching the Quark Model

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  1. Simon Capstick, Florida State University

  2. Unquenching the Quark Model • Motivation • Required ingredients • Self-consistent baryon masses in presence of continuum (baryon-meson) states • SU(6) (flavor-spin) symmetry • Minimum set of intermediate states • Prior and ongoing work • Lessons from covariant calculation of - • Collaborators: Danielle Morel (PhD@FSU, JMU); Mike Pichowsky (Chicago mercantile exchange), Sameer Walawalkar (PhD@FSU, CMU);Eric Swanson (Pittsburgh);Ted Barnes (ORNL/U Tennesssee, Knoxville) Simon Capstick, Florida State University

  3. Motivation E. Teller: “From mesons all manner of forces you get; the infinite part, you simply forget” Simon Capstick, Florida State University

  4. Motivation… • In QCD qqq(qq) configurations possible in baryons: effect on CQM? • Expand in basis of baryon-meson intermediate states • B’M loops a self energies ~O(G), mixing • requires calculation of form factors • Many thresholds close in energy (k) Simon Capstick, Florida State University

  5. Required ingredients To calculate self energies and mixing: • Need model of spectrum (including states not seen in experiment) a wave functions a vertices • Need model of B(0) a B’(-k)M(k) vertices and their momentum dependence • Focus on baryons • Many calculations of effects in mesons exist, not surveyed here Simon Capstick, Florida State University

  6. Self-consistent mass calculation • Assume your qqq (bare) quark model baryon masses M0B depend on: • Parameters: strong coupling 0, quark mass m0, string tension b0…, 3P0 (pair creation) coupling strength  • Hamiltonians: Hqqq, Hpc • Parameters determined by a fit to the spectrum and decays in the absence of higher Fock-space components EB = M0B(0,m0,b0,…) = physical mass Simon Capstick, Florida State University

  7. Self-consistent quark model masses… • Correction due to B ! B’C iswhere • Imaginary part of loop integral is ½B! B’C • Perform sum over B’C intermediate states • Adjust parameters 0,m0,b0… for self-consistent solution with EB = physical mass • In principle should solve a similar equation for meson masses EC Simon Capstick, Florida State University

  8. Self-consistent mass calculation • Equivalent to second order perturbation theory in decay Hamiltonian Hpc • Can calculate the (momentum space) continuum (B’C) component of physical baryon states • Can calculate the mixing between different baryon states due to continuum states (B! B’C! B’’) Simon Capstick, Florida State University

  9. Intermediate (continuum) states • Baryon self energy due to individual B’M loops are comparable to widths – convergence? • Calculations applied to N, D, L, S, ground and L=1 states; • Intermediate states B’M • Ground states baryonsB’{N, D, L, S, S*} • Mesons: • Blask, Huber and Metsch (’87)used ground state (pseudoscalar) octet mesonsM{p, K, h , h} • Zenczykowski (’86), Silvestre-Brac and Gignoux (’91), Y. Fujiwara (’93) used pseudoscalar and vector mesonsM{p, K, h, h, r, w, K*} Simon Capstick, Florida State University

  10. Intermediate states… • Essential problem: there are lots of B’M thresholds nearby in energy • Nr, Dr similar thresholds to N(1535)p, L(1116)K, etc. • Cannot study effects on spectrum by restricting M to p (or even all pseudoscalars) or B’ to N, D (or even all octet and decuplet ground states) Simon Capstick, Florida State University

  11. SU(6) (flavor-spin) symmetry • Zenczykowski: gedanken experiment • assume SU(3)fSU(2)spin, only ground state B, B’ and M exist • assume all octet and decuplet ground state baryons have same mass M0B & same wvfn. • assume all pseudoscalar and vector ground state mesons have mass M0M & same wvfn. • All loop integrals now the same, apart from SU(6) factor at vertices: D = N ? Simon Capstick, Florida State University

  12. SU(6)flavor-spin weights Simon Capstick, Florida State University

  13. SU(6) flavor symmetry limit aSum of loops for N and D same only if include all B’M combinations (non-strange, strange, or both) consistent with quantum numbers • Need both pseudoscalar and vector mesons • This is true for any baryon in the [56,0+] ground states (octet and decuplet baryons), sum is 48 • Also true in 3P0 model (reduces to SU(6)W in this limit) • If all the thresholds and all wave functions are the same, there are no mass splittings between ground state baryons • Requires: • SU(3)f breaking turned off (ms = mu,d) • Spin-dependent splittings (-N, -, etc.) turned off Simon Capstick, Florida State University

  14. Away from SU(3)f limit • Relax assumption of SU(3)f symmetry in meson & baryon masses (Tornqvist & Zenczykowski, ’84) • Use • Assume: • Spectral function (s,mB,mM) broad, on scale of baryon mass splittings (ultimately calculate using 3P0) Assume form (s,mB,mM) = (s½ -[mB+mM]) • Assume: m0N = m0´ m00m0 = m0L = m0* = m00 + 0´ m01 m0X = m0X*= m00 + 20´ m02 m0W=m00 + 30´ m03 • Similar pattern of SU(3)f breaking for meson masses (m = m, ideal mixing,…) Simon Capstick, Florida State University

  15. Away from SU(3)f limit… • Only two thresholds for each external baryon with k strange quarks: • Non-strange quark pair production Ek0, strange quark pair production Ek0+2 • Call sums over relevant SU(6) weights wk0, and wk2 • Adopt similar notation for output baryon masses: mN = m´ m0m = mL = m* ´ m1 mX = mX*´ m2 mW´m3 • Mass shifts: Simon Capstick, Florida State University

  16. Away from SU(3)f limit… • Ignores factor • [Close to 1 if mk small c.f. hxi +Ek0, but following works without this assumption] • From SU(6)/3P0: wk+12 - wk2 = 4 for k = 0,1,2 • Subtract integral relation for k from the same one for k+1 • Work to first order in mk+1-mk, 0,  Simon Capstick, Florida State University

  17. Away from SU(3)f limit… • Independent of k: • Output baryon masses show same pattern as input masses • The difference in pole positions in the r.h.s. of the (self-consistent) integral equation leads to  in denominator Simon Capstick, Florida State University

  18. Away from SU(6)fs limit • Now assume: • Ground-state baryon masses exhibit this simple equal-spacing pattern • PS and V mesons have their experimental values • Now output masses exhibit SU(6)fs breaking effects • E.g. m - mN 0 • In first order in meson mass differences, c´ cos(P), s´ sin(P) (perfect mixing P=10o) • Constant C / d(-N)/dEthreshold > 0 (as threshold becomes more distant, mass shift decreases) [but 0.18  + 0.82 0] Simon Capstick, Florida State University

  19. Away from SU(6)fs limit • Similar formulae can be derived for *-, -, *-,... • If intermediate baryon masses already split (by e.g. OGE, with similar pattern) • If meson masses satisfy Gell-Mann Okubo SU(6) mass formula • Get 2N + 2 = 3 +  (Gell-Mann Okubo, octet) • Get * -  = * - * =  - * (equal spacing rule for decuplet) • Get * -  = * -  (SU(6) relation) • Recover SU(6) relations for baryon masses! Simon Capstick, Florida State University

  20. Both mechanisms necessary • Result is very interesting • Can interpret SU(6) breaking effects in baryon spectrum as partly due to spin-flavor dependent exchanges (e.g. OGE) • Partly due to loop effects • (not same as OBE) Simon Capstick, Florida State University

  21. Results from prior work… • Effects on spectrum are substantial • Zenczykowski finds many mass splittings close to analyses without qqq residual interactions • Other calculations show splittings in bare P-wave baryon masses which resemble spin-orbit effects (Blask, Huber and Metsch, Silvestre-Brac and Gignoux, Fujiwara) • Solution to spin-orbit problem in baryons? • Lack self-consistent treatment of external and intermediate states—converged? • Such convergence slower in 3P0 NRQM: mesons, Geiger and Isgur • Faster in covariant model based on Schwinger-Dyson Bethe-Salpeter approach: M. Pichowsky, S. Walawalkar, SC Simon Capstick, Florida State University

  22. Baryon self energies in relativized 3P0 model • PhD thesis of Danielle Morel (PhD@FSU, now at JMU); study N, D ground and L=1 excited external states • Calculate vertices as a function of loop momentum using 3P0 model (analytic, Maple) • Use mixed relativized-model wavefunctions (expanded up to N=7 band) • Include intermediate states BM with • Mesons M{p, K, h, h, r, w, K*} • Baryons B{N, D, L, S, S*}, including all excitations up to N=3 band • Roughly 300 to 500 intermediate states! Simon Capstick, Florida State University

  23. Baryon self energies in relativized 3P0 model… • Usual 3P0 model gives vertices that are too hard, loops get large contributions from high momenta • Soften with pair-creation operator form factor ~exp(-f2[pq-pq]2) • Currently revisiting calculation to allow self-consistent renormalization of quark model parameters (hard work!) • T. Barnes and E. Swansonlooking at shifts in charmonium spectrum due to D,D*,Ds,Ds* meson pairs Simon Capstick, Florida State University

  24. - splitting in a covariant model • Are large self energies from hadron loops an artifact of a non-relativistic approach, or of the 3P0 model? • M. Pichowsky, S. Walawalkar, S.C. [PRD60, 054030 (1999)] • Examine  and  self energies in covariant model based on Schwinger-Dyson approach • Calculated pseudoscalar-pseudoscalar and vector-pseudoscalar loops Simon Capstick, Florida State University

  25. Loop calculation • Vector-pseudoscalar-pseudoscalar (VPP) form factors from quark loop integrations (similarly for VVP) • Requires: • knowledge of the u-, d- and s-quark propagators • P and V meson Bethe-Salpeter amplitudes • taken from phenomenological studies of EM form factors and strong and weak decays of P and V mesons Simon Capstick, Florida State University

  26. Form factors • Form factors are evaluated numerically and fit to simple exponential form (VPP) • fVPP(p1,p2) • Note additional suppression when mP > mV/2 Simon Capstick, Florida State University

  27. Results • Self energies rapidly decrease with increasing mass of the intermediate state mesons • Form factors significantly softer than in NRQM/3P0 model calculations • Additional suppression with mass Simon Capstick, Florida State University

  28. Conclusions/Outlook • The next Fock-space component is likely more important than differences among qqq models • calculating its effects requires: • Use of full SU(6)-related set of intermediate states, spatially-excited intermediate baryons • Careful treatment of mixing • Renormalization of parameters in quark model needs to be carried out • Renormalize s, quark masses, string tension • This requires examining mass shifts of more than just N, , and their negative-parity excitations • Need additional suppression when mB >> mB ’+ mM Simon Capstick, Florida State University

  29. Simon Capstick, Florida State University

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