Download - Unquenching the Quark Model
BRAG meeting@Bonn 9/2/07-1Simon Capstick, Florida State University
BRAG meeting@Bonn 9/2/07-2Simon Capstick, Florida State University
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)
BRAG meeting@Bonn 9/2/07-3Simon Capstick, Florida State University
Motivation
E. Teller:
“From mesons all manner of forces you get; the infinite part, you simply forget”
BRAG meeting@Bonn 9/2/07-4Simon Capstick, Florida State University
Motivation…• In QCD qqq(qq) configurations possible
in baryons: effect on CQM?– Expand in basis of baryon-meson
intermediate states– B’M loops self energies ~O(), mixing – requires calculation of form factors– Many thresholds
close in energy
(k)
BRAG meeting@Bonn 9/2/07-5Simon Capstick, Florida State University
Required ingredients
To calculate self energies and mixing:• Need model of spectrum (including states
not seen in experiment) wave functions vertices
• Need model of B(0) B’(-k)M(k) vertices and their momentum dependence
– Focus on baryons• Many calculations of effects in mesons
exist, not surveyed here
BRAG meeting@Bonn 9/2/07-6Simon Capstick, Florida State University
Self-consistent mass calculation
• Assume your qqq (bare) quark model baryon masses M0
B 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 componentsEB = M0
B (0,m0,b0,…) = physical mass
BRAG meeting@Bonn 9/2/07-7Simon Capstick, Florida State University
Self-consistent quark model masses…
• Correction due to B ! B’C is
where
– Imaginary part of loop integral is ½B! 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
BRAG meeting@Bonn 9/2/07-8Simon Capstick, Florida State University
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’’)
BRAG meeting@Bonn 9/2/07-9Simon Capstick, Florida State University
Intermediate (continuum) states
• Baryon self energy due to individual B’M loops are comparable to widths – convergence?
• Calculations applied to ground and L=1 states;
• Intermediate states B’M– Ground states baryons B’{*} – Mesons:
• Blask, Huber and Metsch (’87) used ground state (pseudoscalar) octet mesons M{}
• Zenczykowski (’86), Silvestre-Brac and Gignoux (’91), Y. Fujiwara (’93) used pseudoscalar and vector mesons M{*}
BRAG meeting@Bonn 9/2/07-10Simon Capstick, Florida State University
Intermediate states…
• Essential problem: there are lots of B’M thresholds nearby in energy– N, similar thresholds to N(1535), (1116)K, etc.
– Cannot study effects on spectrum by restricting M to (or even all pseudoscalars) or B’ to N, (or even all octet and decuplet ground states)
BRAG meeting@Bonn 9/2/07-11Simon Capstick, Florida State University
SU(6) (flavor-spin) symmetry• Zenczykowski: gedanken
experiment• assume SU(3)fSU(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: = N ?
BRAG meeting@Bonn 9/2/07-12Simon Capstick, Florida State University
SU(6)flavor-spin weights
N K K * K N N0 0
N 25/6 32/6 1/6 9/6 8/6 (c-2½s)2
/6(2½c+s)2
/6
0 0
8/6 25/6 8/6 0 10/6 0 0 5(c-2½s)2
/65(2½c+s)2
/6
N K* K* *K* N SUM
N 59/6 64/6 11/6 27/6 16/6 33/6 0 48
16/6 95/6 16/6 0 38/6 0 57/6 48
BRAG meeting@Bonn 9/2/07-13Simon Capstick, Florida State University
SU(6) flavor symmetry limit
Sum of loops for N and 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 3P
0 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
BRAG meeting@Bonn 9/2/07-14Simon Capstick, Florida State University
Away from SU(3)f limit• Relax assumption of SU(3)f symmetry in meson &
baryon masses (Tornqvist & Zenczykowski, ’84)
– Use – Assume:
1. Spectral function(s,mB,mM) broad, on scale of baryon mass splittings (ultimately calculate using 3P0)Assume form (s,mB,mM) = (s½ -[mB+mM])
2. Assume: m0N = m0
´ m00
m0 = m0
= m0* = m0
0 + 0 ´ m01
m0 = m0
* = m00 + 20 ´ m0
2 m0
= m00 + 30 ´ m0
3
3. Similar pattern of SU(3)f breaking for meson masses (m= m, ideal mixing,…)
BRAG meeting@Bonn 9/2/07-15Simon Capstick, Florida State University
Away from SU(3)f limit…
• Only two thresholds for each external baryon with k strange quarks:– Non-strange quark pair production Ek
0, strange quark pair production Ek
0+2• Call sums over relevant SU(6) weights wk
0, and wk
2
• Adopt similar notation for output baryon masses: mN = m ´ m0
m = m = m* ´ m1 m = m* ´ m2 m ´ m3
• Mass shifts:
BRAG meeting@Bonn 9/2/07-16Simon Capstick, Florida State University
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 - wk
2 = 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,
BRAG meeting@Bonn 9/2/07-17Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-18Simon Capstick, Florida State University
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]
BRAG meeting@Bonn 9/2/07-19Simon Capstick, Florida State University
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!
BRAG meeting@Bonn 9/2/07-20Simon Capstick, Florida State University
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)
BRAG meeting@Bonn 9/2/07-21Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-22Simon Capstick, Florida State University
Baryon self energies in relativized 3P0 model
• PhD thesis of Danielle Morel (PhD@FSU, now at JMU); study 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{*} • Baryons B{*}, including all
excitations up to N=3 band• Roughly 300 to 500 intermediate states!
BRAG meeting@Bonn 9/2/07-23Simon Capstick, Florida State University
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. Swanson looking at shifts in charmonium spectrum due to D,D*,Ds,Ds
* meson pairs
BRAG meeting@Bonn 9/2/07-24Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-25Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-26Simon Capstick, Florida State University
Form factors
• Form factors are evaluated numerically and fit to simple exponential form (VPP)– fVPP(p1,p2)
• Note additional suppression when mP > mV/2
BRAG meeting@Bonn 9/2/07-27Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-28Simon Capstick, Florida State University
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
BRAG meeting@Bonn 9/2/07-29Simon Capstick, Florida State University